As early as you reasonably can. Medical school admissions are rolling, which means schools review applications and fill interview seats as they arrive, not all at once after a deadline. The exact same application is meaningfully stronger submitted in June than in September.
Aim to submit your AMCAS or AACOMAS primary within the first couple of weeks after it opens in late spring, and turn each secondary essay around within about two weeks of receiving it. Being early and complete is one of the largest advantages fully within your control, so protect it by pre-writing your personal statement and your secondary essays before the cycle even opens.