Prompted by peculiar spectroscopic variability observed in SDSS/APOGEE $H$-band spectra, we monitored the Be star HD 55606 using optical spectroscopy and found that it is an exotic double-lined spectroscopic binary (SB2) consisting of a Be star and a hot, compact companion that is probably an OB subdwarf (sdOB) star. Motion of the sdOB star is traced by its impact on the strong He~I lines, observed as radial velocity ($V_{r}$) variable, double-peaked emission profiles with narrow central absorption cores. Weak He II 4686 {AA} absorption associated with the companion star is detected in most spectra. Use of the emission peaks of low-ionization emission lines to trace the Be star $V_{r}$ and the He I lines to trace the companion star $V_{r}$ yields a circular orbital solution with a 93.8-day period and masses of $M_{rm Be}=6.2$ $M_{rm odot}$ and $M_{rm sdOB}=0.9$ $M_{rm odot}$ in the case of $i=80^{circ}$. HD 55606 exhibits a variety of phase-locked variability, including the development of shell lines twice per orbit. The shell phases coincide with variation in the double emission peak separations, and both forms of variability are likely caused by a two-armed spiral density perturbation in the Be disk. The intensity ratios of the double emission peaks are also phase-locked, possibly indicating heating by the sdOB star of the side of the Be disk facing it. HD 55606 is a new member of the growing sample of Be+sdOB binaries, in which the Be stars rapid rotation and ability to form a disk can be attributed to past mass transfer.
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