Quarks and gluons are the fundamental building blocks of protons and neutrons, the constituents of most of the visible matter in the universe. On the one hand, quantum chromodynamics (QCD), the gauge theory of the strong force, describes our current understanding of quarks and gluons. On the other, parton distribution functions (PDFs) capture the simplest aspects of nucleon structure and are one of the primary theoretical tools relating nucleons to their constituents. Systematically determining PDFs from QCD, however, has proved challenging. In this talk I propose a framework that overcomes the problem of power-divergent mixing, which has so far prevented direct determination of nucleon structure from lattice QCD. I will explore this framework using the toy model of scalar field theory and briefly discuss potential applications beyond simple nucleon structure.