  We use the variational principle approach to derive the large $N$ holographic
dictionary for two-dimensional $T\bar T$-deformed CFTs, for both signs of the
deformation parameter. The resulting dual gravitational theory has mixed
boundary conditions for the non-dynamical graviton; the boundary conditions for
matter fields are undeformed. When the matter fields are turned off and the
deformation parameter is negative, the mixed boundary conditions for the metric
at infinity can be reinterpreted on-shell as Dirichlet boundary conditions at
finite bulk radius, in agreement with a previous proposal by McGough, Mezei and
Verlinde. The holographic stress tensor of the deformed CFT is fixed by the
variational principle, and in pure gravity it coincides with the Brown-York
stress tensor on the radial bulk slice with a particular cosmological constant
counterterm contribution. In presence of matter fields, the connection between
the mixed boundary conditions and the radial "bulk cutoff" is lost. Only the
former correctly reproduce the energy of the bulk configuration, as expected
from the fact that a universal formula for the deformed energy can only depend
on the universal asymptotics of the bulk solution, rather than the details of
its interior. The asymptotic symmetry group associated with the mixed boundary
conditions consists of two commuting copies of a state-dependent Virasoro
algebra, with the same central extension as in the original CFT.
