What exactly should be the game plan here for black (me) to breakthrough in this position?
This position is even. Bf6 should go to e5. There is no real breakthrough for either side. The plan is not to make a mistake.
You first want to contain the kingside expansion by white before you are squeezed for space - so you need to gain control over the important dark squares, namely f4. That's why the centralising move 19... Be5 comes immediately to mind. You don't have time to wait, (otherwise white will play 20.Qe2 and then 21. f4 uncontested), and it also allows you to bring the queen into the game to help with the defense.
I agree with tpr.
If you just hate draws regardless of whether you're really improving your position, you could try Bh4 --> Qf6 (Qxf6 gxf6) and then advance with the king.
You don't have any realistic breakthrough. White is the only one with an active plan in this position.
I would have tried ...Be5 and ...g5 to gain control of f4.
Maybe after controlling the f4 square play on the h-file could be possible if white plays bad, otherwise a draw
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