While monotone boards require caution, we must value bet our improved hands and flushes rather than checking back out of fear.
Checking is the preferred play here. On a monotone Ace-high board, our range advantage is thin, and our specific hand has enough showdown value to check back.
Note: Betting into a monotone board with middle pair is risky; checking back realizes equity and protects our range.
Once we turn a set, we must bet for value. Even though the flush is possible, we have massive equity and need to charge draws and Ax hands.
Checking back a flush on the river is a significant value loss. We have the Ts, which is a strong flush, and BB's range is capped after calling two streets.
Note: Checking back a flush on a 4-flush board is a massive missed value opportunity against Ax or lower spade holdings.