  The quantization of pure 3D gravity with Dirichlet boundary conditions on a
finite boundary is of interest both as a model of quantum gravity in which one
can compute quantities which are "more local" than S-matrices or asymptotic
boundary correlators, and for its proposed holographic duality to $T
\overline{T}$-deformed CFTs. In this work we apply covariant phase space
methods to deduce the Poisson bracket algebra of boundary observables. The
result is a one-parameter nonlinear deformation of the usual Virasoro algebra
of asymptotically AdS$_3$ gravity. This algebra should be obeyed by the stress
tensor in any $T\overline{T}$-deformed holographic CFT. We next initiate
quantization of this system within the general framework of coadjoint orbits,
obtaining - in perturbation theory - a deformed version of the
Alekseev-Shatashvili symplectic form and its associated geometric action. The
resulting energy spectrum is consistent with the expected spectrum of
$T\overline{T}$-deformed theories, although we only carry out the explicit
comparison to $\mathcal{O}(1/\sqrt{c})$ in the $1/c$ expansion.
