We reformulate the Palatini action for general relativity in terms of moving frames that exhibit local Galilean covariance in a large speed of light expansion. For this, we express the action in terms of variables that are adapted to a Galilean subgroup of the GL(n, R) structure group of a general frame bundle. This leads to a novel Palatini-type formulation of general relativity that provides a natural starting point for a first-order nonrelativistic expansion. in doing so, we show how a comparison of Lorentzian and Newton-Cartan metric compatibility explains the appearance of torsion in the nonrelativistic expansion.