  We report theoretical results for heat transport by quasiparticle excitations
in superfluid $^3$He infused into silica aerogel engineered with uniaxial
anisotropy. For this system two distinct equal spin pairing (ESP) superfluid
phases have been reported based on NMR spectroscopy. Theoretical analysis
predicts the first ESP state to be the chiral A phase with chiral axis aligned
along the strain axis, and the lower temperature phase to be a polar-distorted
chiral phase with random transverse fluctuations in the orientation of the
chiral axis. We report calculations of heat transport for the high temperature
chiral phase, including an anomalous (zero field) thermal Hall current
originating from branch conversion scattering of Bogoliubov quasiparticles by
the chiral order parameter induced by potential scattering by the silica
aerogel. Observation of an anomalous thermal Hall current would provide a
direct signature of the underlying chirality and topology of the superfluid
phase of $^3$He in "stretched" silica aerogels.
