You probably have to be recommended for promotion before you can take the test.
Quite possible. But "Thine Own Self" gave us the impression that Crusher, eight years ago, personally acted on a desire to get promoted: the whole point was that she didn't wait for, and didn't have to wait for, a recommendation. Had she just waited, she could have stayed a LtCmdr forever and supposedly still been qualified to be CMO to giant starships, including the E-D.
Whether Crusher's promotion to Commander was in any way related to the fact that she also liked to stand bridge watches, we don't know. It actually sounds pretty unlikely.
Now, Troi was explicit about wanting to do the BOT and gain the rank of Commander. But we have no idea whether the latter would automatically follow from the former. Whether the former would be required for the latter, the case of Pulaski would seem to establish "no, definitely not".
What Troi wanted need not have been exactly the same thing Crusher had already done. Certainly Troi doesn't make the claim that this would be the same thing.
Timo Saloniemi