I saw Akira when it initially came out in the cinema (showing my age here). I believe it was the first "adult oriented" anime ever given a cinema release in my country.
I thought it was incredibly dissapointing - a good concept and initial premise but with badly-written female characters (something which I was to later learn was the standard for anime in general), a stupid ending and about half an hour too long. In fact if the last 30 minutes of the film was simply removed altogether with no other changes I believe Akira would be a far superior film.
I didn't actually "enjoy" an anime until I saw Spirited Away many years later. I was hesitant about seeing it at all, having endured travesties of extreme wasted potential and pretentiousness such as Ghost In The Shell, Neon Genesis Evangelion, Battle Angel Atilla etc etc at the insistence of friends, but they practically begged me on hands and knees to check out Spirited Away so I agreed. I was pleasantly surprised to find that Miyazaki broke nearly all the standard anime conventions thus improving on the formula dramatically, and most unexpectedly of all, actually took the time to animate the characters smoothly. So in terms of the criteria the OP laid down (i.e that it actually is of good quality), I guess that Spirited Away would be my first mature anime although in actual fact it was about my 40th.