  The kinetic Sunyaev Zel'dovich (kSZ) effect is a blackbody cosmic microwave
background (CMB) temperature anisotropy induced by Thomson scattering off free
electrons in bulk motion with respect to the CMB rest frame. The statistically
anisotropic cross-correlation between the CMB and galaxy surveys encodes the
radial bulk velocity (more generally, the remote dipole field), which can be
efficiently reconstructed using a quadratic estimator. Here, we develop and
implement a quadratic estimator for the remote dipole field to data from the
Planck satellite and the unWISE galaxy redshift catalog. With this data
combination, we forecast a $\sim 1$-$\sigma$ detection within $\Lambda$CDM
assuming a simple model for the distribution of free electrons. Using
reconstructions based on individual frequency temperature maps, we characterize
the impact of foregrounds, concluding that they can be effectively mitigated by
masking and removing the estimator monopole. We demonstrate that
reconstructions based on component-separated CMB maps have no detectable biases
from foregrounds or systematics at the level of the expected statistical error.
We use these reconstructions to constrain the multiplicative optical depth bias
to $b_v < 1.40$ at $68 \%$ confidence. Our fiducial signal model with $b_v =1$
is consistent with this measurement. Our results support an optimistic future
for kSZ velocity reconstruction with near-term datasets.
