  This paper provides an asymptotic description of a solution to the
Burgers-Hilbert equation in a neighborhood of a point where two shocks
interact. The solution is obtained as the sum of a function with $H^2$
regularity away from the shocks plus a corrector term having an asymptotic
behavior like |x|ln|x| close to each shock. A key step in the analysis is the
construction of piecewise smooth solutions with a single shock for a general
class of initial data.
