The thermal stability of proteins can be altered when they interact with small molecules, other biomolecules or are subject to post-translation modifications. Thus monitoring the thermal stability of proteins under various cellular perturbations can provide insights into protein function, as well as potentially determine drug targets and off-targets. Thermal proteome profiling is a highly multiplexed mass-spectrommetry method for monitoring the melting behaviour of thousands of proteins in a single experiment. In essence, thermal proteome profiling assumes that proteins denature upon heating and hence become insoluble. Thus, by tracking the relative solubility of proteins at sequentially increasing temperatures, one can report on the thermal stability of a protein. Standard thermodynamics predicts a sigmoidal relationship between temperature and relative solubility and this is the basis of current robust statistical procedures. However, current methods do not model deviations from this behaviour and they do not quantify uncertainty in the melting profiles. To overcome these challenges, we propose the application of Bayesian functional data analysis tools which allow complex temperature-solubility behaviours. Our methods have improved sensitivity over the state-of-the art, identify new drug-protein associations and have less restrictive assumptions than current approaches. Our methods allows for comprehensive analysis of proteins that deviate from the predicted sigmoid behaviour and we uncover potentially biphasic phenomena with a series of published datasets.