  The interest in Fe-chalcogenide unconventional superconductors is intense
after the critical temperature of FeSe was reported enhanced by more than one
order of magnitude in the monolayer limit at the interface to an insulating
oxide substrate. In heterostructures comprising interfaces of FeSe with
topological insulators, additional interesting physical phenomena is predicted
to arise e.g. in form of {\it topological superconductivity}. So far
superconductive properties of Fe-chalcogenide monolayers were mostly studied by
local scanning tunneling spectroscopy experiments, which can detect pseudo-gaps
in the density of states as an indicator for Cooper pairing. Direct macroscopic
transport properties which can prove or falsify a superconducting phase were
rarely reported due to the difficulty to grow films with homogeneous material
properties. Here we report on a promising growth method to fabricate continuous
carpets of monolayer thick FeSe on molecular beam epitaxy grown Bi$_2$Se$_3$
topological insulator thin films. In contrast to previous works using
atomically flat cleaved bulk Bi$_2$Se$_3$ crystal surfaces we observe a strong
influence of the high step-edge density (terrace width about 10~nm) on
MBE-grown Bi$_2$Se$_3$ substrates, which significantly promotes the growth of
coalescing FeSe domains with small tetragonal crystal distortion without
compromising the underlying Bi$_2$Se$_3$ crystal structure.
