  Motivated by experiments on the carrier-doped bilayer iridate
$(\text{Sr}_{1-x}\text{La}_x)_3\text{Ir}_2\text{O}_7$, we study the dynamics of
a single doped electron in a bilayer magnet in the presence of spin-orbit
coupling, taking into account the spatially staggered rotation of IrO$_6$
octahedra. We employ an effective single-orbital bilayer $t$-$J$ model,
concentrating on the quantum paramagnetic phase near the magnetic quantum
critical point. We determine the carrier dispersion using a combination of
self-consistent Born and bond-operator techniques. Extrapolating to finite
small carrier density we find that, for experimentally relevant parameters, the
combination of octahedral rotation and spin-orbit coupling induces a band
folding which results in a Fermi surface of small double electron pockets, in
striking agreement with experimental observations. We also determine the
influence of spin-orbit coupling on the location of the quantum critical point
in the undoped case, and discuss aspects of the global phase diagram of doped
bilayer Mott insulators.
