In Borel graph combinatorics, one often produces a structure (e.g. a coloring) by dividing a graph into subgraphs with finite connected components, then defining the structure on those components via some uniformization result. We first give an overview of some recent work formalizing these notions. We then present an application to the problem of edge coloring. For Borel actions of certain groups, we find "degree plus one" Borel edge colorings, matching the classical bound of Vizing. Furthermore, for finitely generated abelian groups, we are able to exactly determine Borel edge chromatic numbers.