  Two-dimensional superconductors have been realized in various atomically thin
films such as the twisted bilayer graphene, some of which are anticipated to
involve unconventional pairing mechanism. Due to their low dimensionality,
experimental probes of the exact nature of superconductivity in these systems
have been limited. We propose, by applying a \emph{vertical} supercurrent to a
bilayer superconductor where the mirror symmetry is naturally broken by the
twisting, there will be anomalous thermal Hall effect induced by the
supercurrent that can serve as a sharp probe for the \emph{in-plane} anisotropy
of the superconducting gap function. This effect occurs in the \emph{absence}
of an external magnetic field and spontaneous breaking of the time-reversal
symmetry in the ground state. We derive explicit formulas for the induced
thermal Hall conductivity and show them to be significant in the examples of
twisted cuprates and twisted FeSe where monolayer superconductivity have
already been observed. Though technical challenges still exist, we propose this
to be a generic probe of the gap anisotropy in a twisted bilayer
superconductor.
