  We introduce a new class of discrete approximations of planar domains that we
call "hedgehog domains". In particular, this class of approximations contains
two-step Aztec diamonds and similar shapes. We show that fluctuations of the
height function of a random dimer tiling on hedgehog discretizations of a
planar domain converge in the scaling limit to the Gaussian Free Field with
Dirichlet boundary conditions. Interestingly enough, in this case the dimer
model coupling function satisfies the same Riemann-type boundary conditions as
fermionic observables in the Ising model. In addition, using the same
factorization of the double-dimer model coupling function as in [17], we show
that in the case of approximations by hedgehog domains the expectation of the
double-dimer height function is harmonic in the scaling limit.
