Early B-type main-sequence (MS) stars (M$_1$ = 5-16 M$_{odot}$) with closely orbiting low-mass stellar companions (q = M$_2$/M$_1$ < 0.25) can evolve to produce Type Ia supernovae, low-mass X-ray binaries, and millisecond pulsars. However, the formation mechanism and intrinsic frequency of such close extreme mass-ratio binaries have been debated, especially considering none have hitherto been detected. Utilizing observations of the Large Magellanic Cloud galaxy conducted by the Optical Gravitational Lensing Experiment, we have discovered a new class of eclipsing binaries in which a luminous B-type MS star irradiates a closely orbiting low-mass pre-MS companion that has not yet fully formed. The primordial pre-MS companions have large radii and discernibly reflect much of the light they intercept from the B-type MS primaries ($Delta$I$_{rm refl}$ = 0.02-0.14 mag). For the 18 definitive MS + pre-MS eclipsing binaries in our sample with good model fits to the observed light curves, we measure short orbital periods P = 3.0-8.5 days, young ages $tau$ = 0.6-8 Myr, and small secondary masses M$_2$ = 0.8-2.4 M$_{odot}$ (q = 0.07-0.36). The majority of these nascent eclipsing binaries are still associated with stellar nurseries, e.g. the system with the deepest eclipse $Delta$I$_1$ = 2.8 mag and youngest age $tau$ = 0.6$pm$0.4 Myr is embedded in the bright H II region 30 Doradus. After correcting for selection effects, we find that (2.0$pm$0.6)% of B-type MS stars have companions with short orbital periods P = 3.0-8.5 days and extreme mass ratios q = 0.06-0.25. This is $approx$10 times greater than that observed for solar-type MS primaries. We discuss how these new eclipsing binaries provide invaluable insights, diagnostics, and challenges for the formation and evolution of stars, binaries, and H II regions.