  The method of variational bootstrapping, based on canonical variational
completion, allows one to construct a Lagrangian for a physical theory
depending on two sets of field variables, starting from a guess of the field
equations for only one such set. This setup is particularly appealing for the
construction of modified theories of gravity since one can take lessons from GR
for the "educated guess" of the metric field equations; the field equations for
the other fields are then fixed by the obtained Lagrangian (up to terms
completely independent from the metric tensor). In the present paper, we
explore the applications of variational bootstrapping in the context of
metric-affine theories of gravity. We find those metric-affine models which
are, in a variational sense, closest to the $\Lambda$CDM model of cosmology.
The method then allows to find "corrected" metric equations and to "bootstrap"
the connection field equations. The Lagrangians obtained via this method,
though imposing some restricting criteria, encompass a wide variety of
metric-affine models. In particular, they allow for quadratic metric-affine
theories only if these avoid non-linear terms in the curvature tensor.
