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In this work, we perform a full-spectrum fitting of 350 massive and passive galaxies selected as cosmic chronometers from the LEGA-C ESO public survey to derive their stellar ages, metallicities, and star-formation histories. We extensively test our results by assessing their dependence on the possible contribution of dust, calibration of noise and signal, and the use of photometric data in addition to spectral information; we as well identify indicators of the correct convergence of the results, including the shape of the posterior distributions, the analysis of specific spectral features, and the correct reproduction of the observed spectrum. We derive a clear age-redshift trend compatible with the aging in a standard cosmological model, showing a clear downsizing pattern, with more massive galaxies being formed at higher redshift ($z_f\sim2.5$) with respect to lower massive ones ($z_f\sim2$). From these data, we measure the differential aging of this population of cosmic chronometers to derive a new measurement of the Hubble parameter, obtaining $H(z=0.8) = 113.1 \pm 15.1 (\mathrm{stat.}) ^{+29.1}_{-11.3} (\mathrm{syst.})\ \mathrm{ km\ s^{-1}\ Mpc^{-1}}$. This analysis allows us for the first time to compare the differential ages of cosmic chronometers measured on the same sample with two completely different methods, the full-spectrum fit (this work) and the analysis of Lick indices, known to correlate with the age and metallicity of the stellar populations \citep{Borghi2022a}. Albeit an understood offset in the absolute ages, the differential ages have proven to be extremely compatible between the two methods, despite the very different data, assumptions, and models considered, demonstrating the robustness of the method.