  We study the AdS/BCFT duality between two-dimensional conformal field
theories with two boundaries and three-dimensional anti-de Sitter space with
two Karch-Randall branes. We compute the entanglement entropy of a bipartition
of the BCFT, on both the gravity side and the field theory side. At finite
temperature this entanglement entropy characterizes the communication between
two braneworld black holes, coupled to each other through a common bath. We
find a Page curve consistent with unitarity. The gravitational result, computed
using double-holographically realized quantum extremal surfaces, matches the
conformal field theory calculation. At zero temperature, we obtain an
interesting extension of the AdS$_3$/BCFT$_2$ correspondence. For a central
charge $c$, we find a gap $(\frac{c}{16},\frac{c}{12})$ in the spectrum of the
scaling dimension $\Delta_{\text{bcc}}$ of the boundary condition changing
operator (which interpolates mismatched boundary conditions on the two
boundaries of the BCFT). Depending on the value of $\Delta_{\text{bcc}}$, the
gravitational dual is either a defect global AdS$_3$ geometry or a single sided
black hole, and in both cases there are two Karch-Randall branes.
