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We utilize 17,000 bright luminous red galaxies (LRGs) from the novel Dark Energy Spectroscopic Instrument Survey Validation spectroscopic sample, leveraging its deep (2.5 hr galaxy-1 exposure time) spectra to characterize the contribution of recently quenched galaxies to the massive galaxy population at 0.4 < z < 1.3. We use Prospector to infer nonparametric star formation histories and identify a significant population of recently quenched galaxies that have joined the quiescent population within the past 1 Gyr. The highest-redshift subset (277 at z > 1) of our sample of recently quenched galaxies represents the largest spectroscopic sample of post-starburst galaxies at that epoch. At 0.4 < z < 0.8, we measure the number density of quiescent LRGs, finding that recently quenched galaxies constitute a growing fraction of the massive galaxy population with increasing look-back time. Finally, we quantify the importance of this population among massive ( log(M⋆/M⊙) > 11.2) LRGs by measuring the fraction of stellar mass each galaxy formed in the gigayear before observation, f 1 Gyr. Although galaxies with f 1 Gyr > 0.1 are rare at z ∼0.4 ( 20.5% of the population), by z ∼0.8, they constitute 3% of massive galaxies. Relaxing this threshold, we find that galaxies with f 1 Gyr > 5% constitute 10% of the massive galaxy population at z ∼0.8. We also identify a small but significant sample of galaxies at z = 1.1-1.3 that formed with f 1 Gyr > 50%, implying that they may be analogs to high-redshift quiescent galaxies that formed on similar timescales. Future analysis of this unprecedented sample promises to illuminate the physical mechanisms that drive the quenching of massive galaxies after cosmic noon.