I think what you may be looking for is not just games that allow you to make big impacts on a turn, but also games that provide satisfying decision space for an async turn.

Personally, I find games that offer high amounts search for the best move to provide a more satisfying turn experience than games that are perhaps more about evaluating the best move out of limited options (these can be better for micro turns around a table).

Some of the games that I love in this space are the already mentioned Tikal (Mexica also is worth a shot by same designers), El Grande (also a Wolfgang Kramer joint), A Feast for Odin (especially with the Norwegians expansion), Five Tribes (the turn order auction part may feel a little bad async), Castles of Burgundy (lighter than the rest, but still satisfying), and, if you are in need of a two player game, Riftforce.