June 1. Applying then gets you first on the desks of the adcoms, meaning you'll be one of the first to get secondaries, making you eligible for an invite to ALL of the interview sessions, and by extension making you competitive for EVERY seat a school has.
This really is completely do-able. By now you should have an idea of who you're going to ask to write letters for you, so just ask them in February or March to beat everyone else who's going to be asking for letters at that time. Take the MCAT in April if you haven't already so you'll get your scores in May. The AMCAS application doesn't change from year to year, so there's no reason that you can't just look at the format they have for this year, make a word file that has all of the info you're going to fill in, and then copy/paste it all in on June 1.
Really, getting it in any time in June and even the first half of July is great for most schools. I just say June because that'll give you extra time to fill out the secondaries, and you really want to be done with all of that by the time the next school year starts.