In order to take full advantage of superior data from upcoming spectroscopic instruments (e.g. MOONS) it is important to develop correspondingly superior analysis techniques, moving beyond index measurements to obtain stronger constraints on subtle physical parameters through full spectral fitting. I will introduce Bagpipes, a new public Python code which allows sophisticated model galaxy spectra to be built and fitted to combinations of spectroscopic and photometric data. Crucially, BAGPIPES allows for the modelling and fitting of systematic uncertainties arising from instrumental and atmospheric effects. I will demonstrate the use of this method to analyse a sample of massive quiescent galaxy spectra from VANDELS at 1.0 < z < 1.3, investigating quenching physics at high redshift by constraining the stellar mass vs stellar age relationship, and probing the relationships between green valley, post-starburst and quiescent galaxies by considering the time evolution of their UVJ colours.