  A magnetic impurity in a fermionic superfluid hosts bound quasiparticle
states known as Yu-Shiba-Rusinov (YSR) states. We argue here that, if the
impurity is mobile (i.e., has a finite mass), the impurity and its bound YSR
quasiparticle move together as a midgap molecule, which has an unusual
"Mexican-hat" dispersion that is tunable via the fermion density. We map out
the impurity dispersion, which consists of an "atomic" branch (in which the
impurity is dressed by quasiparticle pairs) and a "molecular" branch (in which
the impurity binds a quasiparticle). We discuss the experimental realization
and detection of midgap Shiba molecules, focusing on lithium-cesium mixtures,
and comment on the prospects they offer for realizing exotic many-body states.
