  We derive a universal asymptotic formula for generic boundary conditions for
the average value of the bulk-to-boundary and boundary Operator Product
Expansion coefficients of any unitary, compact two-dimensional Boundary CFT
(BCFT) with $c>1$. The asymptotic limit consists of taking one or more boundary
primary operators -- which transform under a single copy of the Virasoro
algebra -- to have parametrically large conformal dimension for fixed central
charge. In particular, we find a \textit{single} universal expression that
interpolates between distinct heavy regimes, exactly as in the case of bulk OPE
asymptotics\cite{Collier:2019weq}. The expression depends universally on the
boundary entropy and the central charge, and not on any other details of the
theory. We derive these asymptotics by studying crossing symmetry of various
correlation functions on higher genus Riemann surfaces with open boundaries.
Essential in the derivation is the use of the irrational versions of the
crossing kernels that relate holomorphic Virasoro blocks in different channels.
Our results strongly suggest an extended version of the Eigenstate
Thermalization Hypothesis for boundary OPE coefficients, where the hierarchy
between the diagonal and non-diagonal term in the ansatz is further controlled
by the boundary entropy. We finally comment on the applications of our results
in the context of $\text{AdS}_3/\text{BCFT}_2$, as well as on the recent
relation of BCFTs with lower dimensional models of evaporating black holes.
