For those not using Beta, this is the method brilliant moves are assigned from what I've seen.
My copy-paste message for people asking about brilliant moves in the chess.com discord:
Here's the explanation behind 99.999% of brilliant moves.
From what I've seen, brilliant moves need to fulfill 3 criteria (at least)
1. The engine only sees the move's continuation after a decently high depth, this doesn't mean the engine can't find it, it means the engine doesn't find it on low depth.
2. The move needs to be the only move in the position that draws or wins.
3. The move needs to have more single continuations after black makes their move. Black can have multiple good continuations, but if after at least one of them white again needs to make an 'only move', it'll at least have a high possibility of it being classified as a brilliant move. You'll find it being classified as 'only moves' in your annotated pgns after a game report, it'll say {critical move}
Because of rule 1 and 3, some moves that we'd call "brilliant moves" aren't brilliant on chess.com. It actually needs to see the continuation to be called brilliant on chess.com. If it's so brilliant that the engine doesn't spot it, it won't count. For those using the old Game report, if you've played a move the engine doesn't find but it doesn't say it's brilliant, analyze on a higher depth than 18, it'll say it's brilliant if it correctly evaluates the position at depth 18.
The same thing applies for rule 3, a great "only move" positional idea doesn't always warrant single continuations afterwards, but it could lead further down the line to a better position.
Also, on a side note, since I've seen a lot of people talk about this,
Previous moves made don't matter, it doesn't look at any history, just at the current position and the move made. It may or may not check the local history of assigning brilliant moves (so it doesn't give 3 brilliant moves for a dumb repitition), but I haven't found enough evidence for that yet.
I was playing against a 1400 bot for fun and this happened:
FEN: 8/8/5p2/5Pp1/8/4K3/8/6kB b - - 0 56