  The study of Higgs boson production at large transverse momentum is one of
the new frontiers for the LHC Higgs physics programme. This paper considers
boosted Higgs production in the Standard Model Effective Field Theory (SMEFT).
We focus on the gluon fusion and $t{\bar t}H$ production processes and study
the effects of three dimension-6 operators: the top Yukawa operator, the
gluon-Higgs effective coupling and the chromomagnetic dipole operator of the
top quark. We perform a detailed study of the sensitivity of current and future
LHC data to the corresponding Wilson coefficients, consistently accounting for
their renormalisation group evolution. We compare the sensitivities obtained
with only linear and linear + quadratic terms in the SMEFT by using the
spectrum shape and the addition of the Higgs signal yields. We also consider
fits of $p_T$ spectra in models with heavy-top partners and in MSSM scenarios
with a light scalar top and study the validity of the SMEFT assumptions as a
function of the new-particle masses and the Higgs $p_T$ range. Finally, we
extract constraints on the Wilson coefficients for gluon fusion from a
simultaneous fit to the ATLAS and CMS data and compare our results with those
obtained in global SMEFT analyses.
