  The inclusive Higgs production rate through gluon fusion has been measured to
be in agreement with the Standard Model (SM). We show that even if the
inclusive Higgs production rate is very SM-like, a precise determination of the
boosted Higgs transverse momentum shape offers the opportunity to see effects
of natural new physics. These measurements are generically motivated by
effective field theory arguments and specifically in extensions of the SM with
a natural weak scale, like composite Higgs models and natural supersymmetry. We
show in detail how a measurement at high transverse momentum of $H\to
2\ell+\mathbf{p}\!\!/_T$ via $H\to \tau\tau$ and $H\to WW^*$ could be performed
and demonstrate that it offers a compelling alternative to the $t\bar t H$
channel. We discuss the sensitivity to new physics in the most challenging
scenario of an exactly SM-like inclusive Higgs cross-section.
