  Minimally modified gravity theories are modifications of general relativity
with two local gravitational degrees of freedom in four dimensions. Their
construction relies on the breaking of 4D diffeomorphism invariance keeping
however the symmetry under 3D diffeomorphisms. Here, we construct these
theories from a Hamiltonian point of view. We start with the phase space of
general relativity in the ADM formalism. Then, we find the conditions that the
Hamiltonian must satisfy for the theory to propagate (up to) two gravitational
degrees of freedom with the assumptions that the lapse and the shift are not
dynamical, and the theory remains invariant under 3D diffeomorphisms. This
construction enables us to recover the well-known "cuscuton" class of
scalar-tensor theories in the unitary gauge. We also exhibit a new class of
interesting theories, that we dubb $f({\cal H})$ theories, where the usual
Hamiltonian constraint $\cal H$ of general relativity is replaced by $f({\cal
H})$ where $f$ is an arbitrary function.
