  In many unconventional superconductors, the presence of a pseudogap - a
suppression in the electronic density of states extending above the critical
temperature - has been a long-standing mystery. Here, we employ combined
\textit{in situ} electrical transport and angle-resolved photoemission
spectroscopy (ARPES) measurements to reveal an unprecedentedly large pseudogap
regime in single-layer FeSe/SrTiO$_3$, an interfacial superconductor where
incoherent Cooper pairs are initially formed above $T_{\Delta}$ $\approx$ 60 K,
but where a zero resistance state is only achieved below $T_{0}$ $<$ 30 K. We
show that this behavior is accompanied by distinct transport signatures of
two-dimensional phase fluctuating superconductivity, suggesting a mixed vortex
state hosting incoherent Cooper pairs which persist well above the maximum
clean limit $T_{c}$ of $\approx$ 40 K. Our work establishes the critical role
of reduced dimensionality in driving the complex interplay between Cooper
pairing and phase coherence in two-dimensional high-$T_c$ superconductors,
providing a paradigm for understanding and engineering higher-$T_{c}$
interfacial superconductors.
