  The Higgs production and decay rates offer a new way to probe new physics
beyond the Standard Model. While dynamics aiming at alleviating the hierarchy
problem generically predict deviations in the Higgs rates, the current
experimental analyses cannot resolve the long- and short-distance contributions
to the gluon fusion process and thus cannot access directly the coupling
between the Higgs and the top quark. We investigate the production of a boosted
Higgs in association with a high-transverse momentum jet as an alternative to
the $t\bar{t}h$ channel to pin down this crucial coupling. Presented first in
the context of an effective field theory, our analysis is then applied to
models of partial compositeness at the TeV scale and of natural supersymmetry.
