Method #1 is to let you select several, trash them, and then select again etc.
Method #2 is like method #1 but also: when you trash, you trigger them one by one.
I don't really see the big upside for creating a special interface for these. I would just literally do what it says, trashing (and confirming) one card at a time. (Of course, I'm a stickler.) There should be a button "Don't trash". When you click it, Monastery is done trashing.
Problems with the two methods:
* Potential edge cases as Navical laid out. With method #1 these can be solved by never allowing you to select more than one card at a time that could potentially trigger something (including Catacombs, Cultists, etc and Inherited Catacombs, Cultists etc). It would make the UI pretty awkward though. And it could be that to make it smoother, you'd have to differentiate between different effects and triggers, like Market Square, Catacombs, Rocks... With method #2, you would have to ask the user which to trigger first, and then it could be that you have to cancel the trashing of an already-selected card.
* I only played woth Monastery once, and as mentioned I made a mistake because of it. Since I knew how to play with it, I excepted to confirm after each card, and so ended up trashing just 1, and didn't understand what was going on in time to undo. This is a problem the way it is now, and also with both methods.
* With both methods there's the problem that it seems like you trash all at once, like with Chapel. That confusion is not very friendly in terms of UI either.
Regarding my suggestion:
* I would think that in the vast majority of cases, you trash 1 or 2 cards with Monastery, and rarely more than 3. If you do trash more, that would probably be like once in the game. Doctor overpay already has you deal with each single card - although you don't need to confirm there. If you're adverse to confirming on trash though, I expect you would have already unchecked that setting.
* It's probably the solution that required the least amount of work (except keeping it like it is of course), so resources can be spent on other things.
Keeping it as-is has the benefit of the least amounf of work and it works smoothly if we say that online Monastery is different than IRL Monastery (which would be the only Dominion card like that except maybe Black Market, but playing BM still works the same). For people who expect IRL Monastery though, it could cause mistakes until they learn the difference.