In Thiemann's derivation of the Ashtekar variables he first enlarges the phase space of the Palatini action, spanning this larger space with canonical variables K and E, K will go away but E will remain in the Ashtekar variables. He shows that the new (K,E) coincide with the Palatini (p,q) variables when a constraint is satisfied; this constraint is satisfied identically in the Palatini geometry. Only then is the connection A introduced, and it replaces the nonce variable K, and the new variables (A,E) are canonical and span the big phase space.
In general is it really true that a connection by itself specifies a geometry? Recall that in traditional Riemann you have first a metric  specified by a symmetric tensor, which restricts your choice of geometries, and then define the connection as a function of your metric (through the Christoffel symbols). This then gives you the curvature tensor and all the rest. But the contribution of the symmetric metric was important.
