  A precise measurement of the Higgs $h \to Z \gamma$ decay is very challenging
at the LHC, due to the very low %SM $h \to Z \gamma\, (Z \to \ell \ell)$
branching fraction and the shortage of kinematic handles to suppress the large
SM $Z \gamma$ background. We show how such a measurement would be significantly
improved by considering Higgs production in association with a hard jet. We
compare the prospective HL-LHC sensitivity in this channel with other Higgs
production modes where $h$ is fairly boosted, e.g.~weak boson fusion, and also
to the potential $h \to Z \gamma$ measurement achievable with a future
$e^{+}e^{-}$ circular collider ({\sl fcc-ee}). Finally, we discuss new physics
implications of a precision measurement of $h \to Z \gamma$.

High Energy Physics - Phenomenology