  The sensitive dependence of monolayer materials on their environment often
gives rise to unexpected properties. It was recently demonstrated that
monolayer FeSe on a SrTiO$_3$ substrate exhibits a much higher superconducting
critical temperature T$_C$ than the bulk material. Here, we examine the
interfacial structure of FeSe / SrTiO$_3$ and the effect of an interfacial
Ti$_{1+x}$O$_2$ layer on the increased T$_C$ using a combination of scanning
transmission electron microscopy and density functional theory. We find
Ti$_{1+x}$O$_2$ forms its own quasi-two-dimensional layer, bonding to both the
substrate and the FeSe film by van der Waals interactions. The excess Ti in
this layer electron-dopes the FeSe monolayer in agreement with experimental
observations. Moreover, the interfacial layer introduces symmetry-breaking
distortions in the FeSe film that favor a T$_C$ increase. These results suggest
that this common substrate may be functionalized to modify the electronic
structure of a variety of thin films and monolayers.
