  A dome-shaped phase diagram of superconducting critical temperature upon
doping is often considered as a hallmark of unconventional superconductors.
This behavior, observed in two-dimensional electron gases in
$\mathrm{SrTiO}_3$-based interfaces whose electronic density is controlled by
field effect, has not been explained unambiguously yet. Here, we elaborate a
generic scenario for the superconducting phase diagram of these oxide
interfaces based on Schr\"odinger-Poisson numerical simulations of the quantum
well and transport experiments on a double-gate field-effect device. We propose
that the optimal doping point of maximum $T_c$ marks the transition between a
single-band and a fragile two-gap s$\pm$-wave superconducting state involving
$t_{2g}$ bands of different orbital character. At the optimal doping point, we
predict and observe experimentally a bifurcation in the dependence of $T_c$ on
the carrier density, which is controlled by the details of the doping
execution. Where applying a back-gate voltage triggers the filling of a
high-energy $d_\mathrm{xy}$ subband and initiates the overdoped regime, doping
with a top-gate delays the filling of the subband and maintains the 2-DEG in
the single-band superconducting state of higher $T_c$.
