  We develop a gluing procedure designed to obtain canonical metrics on
connected sums of Einstein four-manifolds. The main application is an existence
result, using two well-known Einstein manifolds as building blocks: the
Fubini-Study metric on $\mathbb{CP}^2$ and the product metric on $S^2 \times
S^2$. Using these metrics in various gluing configurations, critical metrics
are found on connected sums for a specific Riemannian functional, which depends
on the global geometry of the factors. Furthermore, using certain quotients of
$S^2 \times S^2$ as one of the gluing factors, critical metrics on several
non-simply-connected manifolds are also obtained.
