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We have discovered a new X-ray emitting compact binary that is the likely counterpart to the unassociated Fermi-LAT GeV $γ$-ray source 4FGL J1120.0-2204, the second brightest Fermi source that still remains formally unidentified. Using optical spectroscopy with the SOAR telescope, we have identified a warm ($T_{\textrm{eff}}\sim8500$ K) companion in a 15.1-hr orbit around an unseen primary, which is likely a yet-undiscovered millisecond pulsar. A precise Gaia parallax shows the binary is nearby, at a distance of only $\sim 820$ pc. Unlike the typical "spider" or white dwarf secondaries in short-period millisecond pulsar binaries, our observations suggest the $\sim 0.17\,M_{\odot}$ companion is in an intermediate stage, contracting on the way to becoming an extremely low-mass helium white dwarf (a "pre-ELM" white dwarf). Although the companion is apparently unique among confirmed or candidate millisecond pulsar binaries, we use binary evolution models to show that in $\sim 2$ Gyr, the properties of the binary will match those of several millisecond pulsar-white dwarf binaries with very short ($< 1$ d) orbital periods. This makes 4FGL J1120.0-2204 the first system discovered in the penultimate phase of the millisecond pulsar recycling process.