  We study a family of (possibly non topological) deformations of $BF$ theory
for the Lie algebra obtained by quadratic extension of $\mathfrak{so}(3,1)$ by
an orthogonal module. The resulting theory, called quadratically extended
General Relativity (qeGR), is shown to be classically equivalent to certain
models of gravity with dynamical torsion. The classical equivalence is shown to
promote to a stronger notion of equivalence within the Batalin--Vilkovisky
formalism. In particular, both Palatini--Cartan gravity and a deformation
thereof by a dynamical torsion term, called (quadratic) generalised Holst
theory, are recovered from the standard Batalin--Vilkovisky formulation of qeGR
by elimination of generalised auxiliary fields.
