  The studies of high-temperature superconductors raise a fundamental question:
Can a small Fermi surface phase, which violates the Luttinger theorem, exist
and give rise to superconductivity? Our work provides a positive answer through
a controlled theory based on a bilayer model with strong inter-layer spin-spin
coupling ($J_\perp$) but no inter-layer hopping ($t_\perp$). Then small hole
doping of the rung-singlet insulator with two electrons per rung naturally
leads to small hole pockets with Fermi surface volume per flavor smaller than
the free fermion result by $1/2$ of the Brillouin zone(BZ). We construct a new
t-J model on a bilayer square lattice, so called ESD t-J model and employ a
generalized slave boson theory, which captures this small Fermi surface phase
at small hole doping $x$. This metallic state is an intrinsically strongly
correlated Fermi liquid beyond weak coupling theory, violating the perturbative
Luttinger theorem but consistent with the Oshikawa's non-perturbative proof. We
further show that it transitions into an inter-layer paired $s'$-wave
superconductor at lower temperature through Feshbach resonance with a virtual
Cooper pair, with a surprising doping-induced crossover from
Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) at higher
hole doping levels. This leads to a superconducting dome centered around
$x=0.5$, with the normal state changing from the conventional Fermi liquid in
the $x>0.5$ to the unusual small Fermi surface state in the $x<0.5$ side. Our
theoretical findings including phase diagrams are also confirmed by density
matrix renormalization group (DMRG) simulation in quasi one dimension. Applying
our theoretical framework, we provide a plausible scenario for the recently
found nickelate La$_3$Ni$_2$O$_7$ materials.
