For the character/word 煎, which of the English words "braise", "pan fry", "sauté" are the best way to think of it? In my Memrise Chinese courses 煎 seems to be translated inconsistently and I'm always getting it wrong. This also leads to confusion when I'm looking at menus in real life.

It seems that the dictionaries I generally use are also not consistent with each other.

In any case English speakers who are not into cooking are not always clear on the differences between these terms and cooking methods, which might be part of the reason for inconsistency is the flashcard courses. Even moreseo given that many English speakers are actually imperfect second language speakers and are still active on language learning sites, forums, flashcard sites, etc.

So out of English "braise", "pan fry", and "sauté", which is the best fit for 煎?

Of course languages don't map 1:1 so the Chinese might be "between" two or all of the English words, or might overlap two or all of them.
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.