Are you working from a particular reference? In Einstein-Cartan theory, I believe that the metric and torsion tensor are taken to be independent variables. In this theory, the covariant derivative is
$$ \nabla_\mu V_\nu = \partial_\mu V_\nu - ({\Gamma^\rho}_{\mu\nu} + {K^\rho}_{\mu\nu} )V_\rho,~~~(*)$$
where ##{\gamma^\rho}_{\mu\nu}## is the usual Christoffel symbol expressed in terms of the metric and ##{K^\rho}_{\mu\nu} ## is the contorsion tensor, related to the torsion tensor ##{T^\rho}_{\mu\nu} ## by
$$ K_{\rho\mu\nu} = \frac{1}{2} ( T_{\rho\mu\nu} - T_{\mu\nu\rho} +T_{\nu\rho\mu} ).$$
In this way, ##{\Gamma^\rho}_{\mu\nu}= {\Gamma^\rho}_{\nu\mu}## and all of the torsion is contained in ##{T^\rho}_{\mu\nu}##. Because of the added term in (*) we say that the connection is no longer compatible with the metric.
I believe that you can also write this theory in terms of a vierbein and a spin connection ##{\omega_\mu}^{ab}+{\gamma_\mu}^{ab}##. Here ##{\omega_\mu}^{ab}## is the parti of the spin connection that can be related to the Christoffel symbol (as in the theory without torsion), while ##{\gamma_\mu}^{ab}## contains the torsion via something like
$$ {\gamma_\mu}^{ab} = K_{\rho\sigma \mu} ( e^{a\rho} e^{b\sigma} - e^{a\sigma} e^{b\rho}).$$
Presumably you can take ##e^a_\mu, {\omega_\mu}^{ab} ##, and ##{\gamma_\mu}^{ab}## as independent variables.
I referred to
this review by Shapiro to gather the formulas together, but I don't think he addresses the variational principle directly in this formalism.