  Since the discovery of superconductivity in cuprate materials, the minimal
ingredients for high-T$_c$ superconductivity have been an outstanding puzzle.
Motivated by the recently discovered nickelate bilayer superconductor
La$_3$Ni$_2$O$_3$ under pressure, we study a minimal bilayer model, in which,
as in La$_3$Ni$_2$O$_3$, inter- and intralayer magnetic interactions but no
interlayer hopping are present: a mixed-dimensional (mixD) $t-J$ model. In the
setting of a mixD ladder, we show that the system exhibits a crossover
associated with a Feshbach resonance: from a closed-channel dominated regime of
tightly bound bosonic pairs of holes to an open-channel dominated regime of
spatially more extended Cooper pairs. The crossover can be tuned by varying
doping, or by a nearest-neighbor Coulomb repulsion $V$ that we include in our
model. Using density matrix renormalization group (DMRG) simulations and
analytical descriptions of both regimes, we find that the ground state is a
Luther-Emery liquid, competing with a density wave of tetraparton plaquettes at
commensurate filling $\delta=0.5$ at large repulsion, and exhibits a pairing
dome where binding is facilitated by doping. Our observations can be understood
in terms of pairs of correlated spinon-chargon excitations constituting the
open channel, which are subject to attractive interactions mediated by the
closed channel of tightly bound chargon-chargon pairs. When the closed channel
is lowered in energy by doping or tuning $V$, a Feshbach resonance is realized,
associated with a dome in the binding energy. Our predictions can be directly
tested in state-of-the art quantum simulators, and we argue that the pairing
mechanism we describe may be realized in the nickelate bilayer superconductor
La$_3$Ni$_2$O$_3$.
