For the character/word 煎, which of the English words "braise", "pan fry", "sauté" are the best way to think of it?
The best fit would be "pan fry" or "sear" - cooking with a small amount of fat, typically in a shallow pan. 煎 goes with such foods as steak, fish filet, pancake, eggs (sunny-side-up or over-easy, not scrambled) ...

FYI, saute or stir-fry is "炒" and braise is usually rendered "紅燒" in menus and recipes.