Do galaxies that quenched at early epochs remain passive since then or do they rejuvenate, experiencing further episodes of star formation? We apply a structure spectro-photometric decomposition method to obtain the spectral energy distributions for bulges and disks in a representative sample of massive galaxies at redshift $0.14 < z \leq 1$. This opens the possibility to study the star formation history of each morphological component, a novelty in the characterization of galaxy evolution at this redshift. We find that bulges display a bimodal distribution of mass-weighted ages, i.e., they form in two waves. The first wave of bulges could start to form as early as $z=10$ and evolve passively for as long as 6 Gyr before re-entering the star-forming main sequence at later times, after acquiring a stellar disk. Being very massive and compact systems, which formed through a very intense episode of star formation, we identify first-wave bulges to be the result of a compaction event that occurred at a very high redshift. These results allow extending to late-type galaxies the two-phase formation scenario currently accepted to shape early-type galaxies: first-wave bulges represent a complementary channel for the evolution of the blue and red-nugget systems observed at cosmic noon, which could enter the main sequence and acquire a stellar disk while evolving in massive disk galaxies as we observe them at $z<1$.