We report the discovery of BD+05 4868 Ab, a transiting exoplanet orbiting a bright (V = 10.16) K-dwarf (TIC 466376085) with a period of 1.27 days. Observations from NASA's Transiting Exoplanet Survey Satellite reveal variable transit depths and asymmetric transit profiles that are characteristic of comet-like tails formed by dusty effluents emanating from a disintegrating planet. Unique to BD+05 4868 Ab is the presence of prominent dust tails in both the trailing and leading directions that contribute to the extinction of starlight from the host star. By fitting the observed transit profile and analytically modeling the drift of dust grains within both dust tails, we infer large grain sizes (∼1–10 μm) and a mass-loss rate of 10 M⊕ Gyr‑1, suggestive of a lunar-mass object with a disintegration timescale of only several Myr. The host star is probably older than the Sun and is accompanied by an M-dwarf companion at a projected physical separation of 130 au. The brightness of the host star, combined with the planet's relatively deep transits (0.8%–2.0%), presents BD+05 4868 Ab as a prime target for compositional studies of rocky exoplanets and investigations into the nature of catastrophically evaporating planets.