You are going to absolutely fall in love with Kunihiko Ikuhara. He eats themes for breakfast, breathes out imagery, and sweats subtext. Every anime he has made is worth watching, but the ones you would be most interested are the ones he had a direct hand in the story and development: Revolutionary Girl Utena, Mawaru Penguindrum, Yurikuma Arashii, and Sarazanmai.

Sarazanmai in particular has some very powerful messages about society and how we connect to eachother. I'm not kidding when I say that when I first watched it, almost every single episode strongly effected me emotionally.

The downside to his work is that you have to be familliar with a lot of japanese culture in order to fully understand all of the themes and how they work. For instance, Penguindrum has a number of references to a Japanese story called The Galaxy Express, and you won't understand some of the social criticism if you hadn't heard about the Tokyo subway sarin attack.

Masaaki Yuasa is also pretty good at using themes in his work, but he does things very differently from anyone else. It's probably closer to motif. For you, I'd specifically recommend The Tatami Galaxy; it's perhaps his most relatable piece of work since it deals directly with the journey into adulthood.

Mushi shi

This was the last anime that I really binged, several years ago. I wouldn't describe it as the "prestige TV" of anime but it does have some interesting depth.

They are neither plants nor animals. They differ from other forms of life such as the micro-organisms and the fungi. Instead they resemble the primeval body of life and are generally known as "Mushi". Their existence and appearance are unknown to many and only a limited number of humans are aware of them. Ginko is a "Mushi-shi" who travels around to investigate and find out more about the "Mushi". In the process, he also lends a helping hand to people who face problems with supernatural occurrences which may be related to the "Mushi".

I think this is what you are looking, these are some with great themes and some have deep story wise:

Cowbow Bebop
Samurai Shamploo
Mushishi
Space Dandy
Parasite
Monster
Serial Experiments Lain
Boogiepop wa Warawanai
Berserk
Vampire Hunter D
Ergo Proxy
Ghost In The Shell
3-gatsu no Lion
Kino’s Journey
Made in Abyss
Death Note

"Anime with depth" is kinda vague, but assuming you mean depth as in strong control of it's themes, I really like what Gatchaman Crowds does with the idea of a Sentai (Power Ranger) team kind of going through the motions, beating up the rubber monsters until their new recruit takes a more socially proactive approach to city protection, getting involved in the community and putting effort into fixing the underlying causes of villainous behavior around town.

In the similar vein is Concrete Revolutio, sort of similar to Watchmen in that it's a period piece of a bunch of anime heroes (magical girl, child ghost, forest yokai, devilman, et al) except it takes place in Japan, in 1968 and the world is changing out from under them. This is probably the most "prestige television" of my picks if that's what you're looking for, but it is still very anime.

On a more personal scope, I'm going to round out picks with SSSS.GRIDMAN, which sells itself as a standard "kid who is contacted by an alien robot to merge with him and become big to fight kaiju" story, but unfolds into an exploration of the kind of people who get wrapped up in sending kaiju to a city and defending the city against them. It dragged it's feet a little compared to some of the other shows that are still talked about, but once you're halfway through, it's something else.

Shigurui, from an imdb review :

The content maybe quite distasteful to some but I would quietly urge the casual viewer to please NOT simply take in the graphic violence at face value and instead look at them as symbols to what may unravel the conflict between the two principal antagonists. Unlike most hero worshiping stories out there, this story has no heroes, only villains with a unique sense of morality and justice. These are stories that can be endlessly dissected and with every viewing this new found perspective can serve aid you decipher a little of that vast unknowable known as Japan. The style of the anime is very decadent. And very much like the politics of Japan during the Edo period - was vastly tumultuous. This story serves as an allegory to that time period through the choices its antagonists makes and its consequences they eventually face. The anime has flawlessly taken in the elements, details and nuances from the manga, and taken it to the next level. The greatest achievement in my opinion was the foreshadowing of the events to come were executed to perfection, like a well aged single malt whiskey. These details can only be noted from repeated viewing.

Or the last part of that review : https://youtu.be/VyhRd5UGjaA?t=390

I'm not sure what you mean by "engage with the media" but here some of my takes, assuming you meant "thinking about stuff" :

Ghost in the Shell Stand Alone Complex discusses a lot of societal and philosophical issues, all within a framework of a police procedural.

In particular, the title refer to memetic imitation, and 4chan and reddit weren't even an idea when the show was first released.

The Melancholy of Suzumiya Haruhi is a intelligent deconstruction/reconstruction of the anime tropes of the naughties (or anime in general).

Maybe due to the language barrier, I don't think I've ever found any anime to be particularly "deep".

Ergo Proxy is about a future where people are dependent on androids in many ways, but a virus is going around that gives the androids consciousness, causing them to retaliate against the humans. The plot is weirder than just that, and it is VERY philosophical, maybe a little too much so, but it has a cool story, interesting concepts, and is rewarding overall. The first episode will suck you in quick if you're anything like me.

Steins Gate explores the concept of time travel pretty deeply, and is also very enjoyable and less brooding than Ergo Proxy for sure. It was actually popular enough to get a movie, which doesn't really happen much with more serious anime (aka not moe crap).

Baccano! is about the search for a mystical immortality elixir in 1920's America. It an interesting story line, and from what I understand is the only anime whose dub is better than the sub, due to it being about westerners.

Mononoke is about a demon slaying medicine salesman. Each episode is its own story, and it explore the mythology of feudal Japan.

I feel like I probably don't watch a whole lot of very deep anime. Hope you find some good stuff in the thread!

Haibane Renmei is a bit obvious but it's something that creates a world which feels coherent and solid and weird.

Yeah, I think you must provide a better definition of “depth” otherwise suggestions may not mean much to you...

One Piece has a lot of depth. The first 4 seasons (whatever that means) are on Netflix. It takes a bit of episodes to realize that depth. At first you enjoy the ride, then the story starts to sink in. Here is a good meme about One piece: https://i.imgur.com/Somx4.png

Since you said Mad Men, then my real recommendation would be Uchuu Kyoudai / Space Brothers. It's about two brothers that want to become astronauts. The story is rather realistic and the characters act like real people.

Otherwise some other good ones: Hyouka, Violet Evergarden, Chihayafuru, Attack on Titans(Not the same depth as Mad Men, but really awesome story buildup with story elements planned from the start.).

Perhaps consider the OG Mobile Suit Gundam?

I think I probably have a few shows I can recommend, but I'm new to the term prestige television, have never seen Mad Men, and am also a bit thick in general... so, when you say "encourages [you] to engage deeper with the media", engage with it how? Do you mean thinking about levels of meaning? Puzzling out clues, symbolism, etc?

I'm not sure this is what you're looking for, but one anime I like is The Twelve Kingdoms. It seems (to me, anyway) targeted at older, more mature audiences on account of it being devoid of a lot of cliches, whether in drawing style, cookie-cutter character personalities, commonly-used plot lines, or clear delineation between good guys and bad guys.

I haven't seen it myself yet, but Legend of the Galactic Heroes is on my list and seems like it fits well what you're looking. It's a long 110-episode space opera packed with well-developed characters, political machinations and intrigue, religion, governing, and of course, huge space battles. I've seen multiple comparisons to Game of Thrones (when it was good).

for anime with real meaningful depth, skip anything form the past 15 years. most of the suggestions in this thread are kinda weak.

Real depth? Go try Rose of Versailles from the 1970s. Supremely intense, and every glance, every eyelash, is full of meaning.

Honestly, a lot of anime get surprisingly deep once they established the basics of the story with -sometimes painful- cliches. Where Western stories tend to establish a base line and go with it, anime often keep adding layers throughout the series. A lot of them are surprisingly deep and adult.