  This paper proves the existence of unstable shocks of the Burgers-Hilbert
equation conjectured in arXiv:2006.05568. More precisely, we construct smooth
initial data with finite $H^9$-norm such that the solution in self-similar
coordinates is asymptotic to the first unstable solution to the self-similar
inviscid Burgers equation. The blowup profile is a cusp with H\"older 1/5
continuity with explicit blowup time and location. Unlike the previously
established stable shocks, the initial data cannot be taken in an open set;
instead, we control the two unstable directions by Newton's iteration.
