  We obtain a Palatini-type formulation for the Galilei and Carroll expansions
of general relativity, where the connection is promoted to a variable. Known
versions of these large and small speed of light expansions are derived from
the Einstein-Hilbert action and involve dynamical Newton-Cartan or Carroll
geometry, along with additional gauge fields at subleading orders. The
corresponding Palatini actions that we obtain in this paper are derived from an
appropriate expansion of the Einstein-Palatini action, and the connection
variable reduces to the Galilei- or Carroll-adapted connection on shell. In
particular, we present the Palatini form for the next-to-leading order Galilean
action and recover the known equations of motion. We also compute the
leading-order Palatini-type action for the Carrollian case and show that, while
it depends on the connection variable, it reduces on shell to the known action
of electric Carroll gravity, which only depends on extrinsic curvature.
