There has been a long-standing factor-of-two tension between the observed star formation rate density and the observed stellar mass buildup after $z\sim2$. Recently we have proposed that sophisticated panchromatic SED models can resolve this tension, as these methods infer systematically higher masses and lower star formation rates than standard approaches. In a series of papers we now extend this analysis and present a complete, self-consistent census of galaxy formation over $0.2 < z < 3$ inferred with the \texttt{Prospector} galaxy SED-fitting code. In this work, Paper I, we present the evolution of the galaxy stellar mass function using new mass measurements of $\sim$10$^5$ galaxies in the 3D-HST and COSMOS-2015 surveys. We employ a new methodology to infer the mass function from the observed stellar masses: instead of fitting independent mass functions in a series of fixed redshift intervals, we construct a continuity model that directly fits for the redshift evolution of the mass function. This approach ensures a smooth picture of galaxy assembly and makes use of the full, non-Gaussian uncertainty contours in our stellar mass inferences. The resulting mass function has higher number densities at a fixed stellar mass than almost any other measurement in the literature, largely owing to the older stellar ages inferred by \texttt{Prospector}. The stellar mass density is $\sim$50% higher than previous measurements, with the offset peaking at $z\sim1$. The next two papers in this series will present the new measurements of star-forming main sequence and the cosmic star formation rate density, respectively.

22 pages, 11 figures