In a natural theory of electroweak symmetry breaking, the degrees of freedom that cure the hierarchy problem are expected to affect also the loop-induced couplings of the Higgs to gluons and photons. However, in the inclusive Higgs production rate measured so far at the LHC, the long-distance contribution of the top quark and the short-distance contribution of new physics cannot be disentangled. In this talk I will argue that the pp -> h + jet process at large pT can resolve this degeneracy, and thus provide another handle besides pp -> t tbar h. As concrete examples, I will discuss the cases of stops in supersymmetry and of top partners in composite Higgs models.