Mature fantasy, eh? Well, to name a few... Berserk comes to mind immediately. Really dark. Umineko no Naku Koro ni is a personal favourite of mine, though it's more mystery than action oriented. The Twelve Kingdoms is my top 1 anime, if you feel like some epic adventure kind, though you'll have to watch about 5 eps before it gets good. The rest is so good it easily makes up for the weak beginning. Hellsing is a vampire thing, a classic, though I'd recommend the OVA Hellsing Ultimate rather than the TV series. Rurouni Kenshin, another classic, is awesome up until it deviates from the original manga after the Shishio-arc.
For a starter anime... well, going with a Miyazaki film might not be a bad idea, seeing as they've been succesful on the Western market as well. Personally I've only seen Spirited Away and Howl's Moving Castle, but I think they're both very good.
BTW, Avatar is epic, mature or not. <.<