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© 2022. The Author(s). Published by the American Astronomical Society. We would like to thank the anonymous referee for improving the content of the manuscript. We are grateful to Ignacio Trujillo and Christopher J. Conselice for the useful discussions and comments. L.C. wishes to thank Cristina Cabello for the support provided while this project was devised and Michele Perna for the fruitful discussions. L.C. acknowledges financial support from Comunidad de Madrid under Atraccion de Talento grant 2018-T2/TIC-11612. L.C. and P.G.P.G. acknowledge support from the Spanish Ministerio de Ciencia, Innovación y Universidades through grant PGC2018-093499-B-I00. J.M.A. acknowledges the support of the Viera y Clavijo Senior program funded by ACIISI and ULL. D.C. is a Ramón-Cajal Researcher and supported by the Ministerio de Ciencia, Innovación y Universidades (MICIU/FEDER) under research grant PGC2018-094975-C21. This work has made use of the Rainbow Cosmological Surveys Database, which is operated by the Centro de Astrobiología (CAB/INTA), partnered with the University of California Observatories at Santa Cruz (UCO/Lick, UCSC)., We investigate the assembly history of massive disk galaxies and describe how they shape their morphology through cosmic time. Using SHARDS and HST data, we modeled the surface brightness distribution of 91 massive galaxies at redshift 0.14 < z <= 1 in the wavelength range 0.5-1.6 mu m, deriving the uncontaminated spectral energy distributions of their bulges and disks separately. This spectrophotometric decomposition allows us to compare the stellar population properties of each component in individual galaxies. We find that the majority of massive galaxies (similar to 85%) build inside-out, growing their extended stellar disk around the central spheroid. Some bulges and disks could start forming at similar epochs, but these bulges grow more rapidly than their disks, assembling 80% of their mass in similar to 0.7 and similar to 3.5 Gyr, respectively. Moreover, we infer that both older bulges and older disks are more massive and compact than younger stellar structures. In particular, we find that bulges display a bimodal distribution of mass-weighted ages; i.e., they form in two waves. In contrast, our analysis of the disk components indicates that they form at z similar to 1 for both first- and second-wave bulges. This translates to first-wave bulges taking longer to acquire a stellar disk (5.2 Gyr) compared to second-wave, less compact spheroids (0.7 Gyr). We do not find distinct properties (e.g., mass, star formation timescale, and mass surface density) for the disks in both types of galaxies. We conclude that the bulge mass and compactness mainly regulate the timing of the stellar disk growth, driving the morphological evolution of massive disk galaxies., Ministerio de Ciencia e Innovación (MICINN)/FEDER, Comunidad de Madrid, Depto. de Estructura de la Materia, Física Térmica y Electrónica, Fac. de Ciencias Físicas, TRUE, pub