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 $h to Z gamma$.
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