  Dual supermassive black holes (SMBHs) at $\sim$kpc scales are the progenitor
population of SMBH mergers and play an important role in understanding the
pairing and dynamical evolution of massive black holes in galaxy mergers.
Because of the stringent resolution requirement and the apparent rareness of
these small-separation pairs, there are scarce observational constraints on
this population, with few confirmed dual SMBHs at $<10$kpc separations at
$z>1$. Here we present results from a pilot search for kpc-scale dual quasars
selected with Gaia Data release 2 (DR2) astrometry and followed up with Hubble
Space Telescope (HST) Wide Field Camera 3 dual-band (F475W and F814W) snapshot
imaging. Our targets are quasars primarily selected with the varstrometry
technique, i.e., light centroid jitter caused by asynchronous variability from
both members in an unresolved quasar pair, supplemented by sub-arcsec pairs
already resolved by Gaia DR2. We find an overall high fraction of HST-resolved
pairs among the varstrometry-selected quasars (unresolved in Gaia DR2), $\sim
30-50\%$, increasing toward high redshift ($\sim 60-80\%$ at $z>1.5$). We
discuss the nature of the 43 resolved sub-arcsec pairs based on HST and
supplementary data. A substantial fraction ($\sim 40\%$) of these pairs are
likely physical quasar pairs or gravitationally lensed quasars. We also
discover a triple quasar candidate and a quadruply lensed quasar, which is
among the smallest-separation quadruple lenses. These results provide important
guidelines to improve varstrometry selection and follow-up confirmation of
$\sim$kpc-scale dual SMBHs at high redshift.
