For a compact $K$, a necessary condition for $C(K)$ to have the Controlled Separable Complementation Property is that $K$ be monolithic. In this paper, we prove that when $K$ contains no copy of $[0,\omega^\omega]$ and the set of points which admit a countable neighborhood base is a cofinite subset of $K$, then monolithicity of $K$ is sufficient for $C(K)$ to enjoy the Controlled Separable Complementation Property. We also show that, for this type of compacta $K$, the space $C(K)$ is separably extensible.
"The controlled separable complementation property and monolithic compacta." Banach J. Math. Anal. 8 (2) 67 - 78, 2014. https://doi.org/10.15352/bjma/1396640052