There is a really nice proof of Borel–Weil–Bott, essentially using parabolic induction (see Proof of Borel-Weil-Bott Theorem, Lurie - A proof of the Borel–Weil–Bott theorem or Demazure - A very simple proof of Bott's theorem). Also, presumably there is a proof if you take the BGG resolution as given, and compare to the Weyl character formula.
However, because of the philosophy that it's worth knowing many proofs of a really fundamental result, I want to ask the
Question: What is your favourite (fundamentally different to the above) proof of Borel–Weil–Bott?