Weight of a topological space is the minimal cardinality of a basis of the topology. A space $X$ is extremely disconnected if open sets in $X$ have open closures.
Is there an example in ZFC of a compact, Hausdorff, extremely disconnected space of weight $\omega_1$?
No. Weight $\kappa$ of any compact, extremely disconnected space satisfies $\kappa^\omega=\kappa$. This is Corollary 6 in:
B.A. Efimov, Extremally disconnected bicompacta and absolutes, Trudy Mosc. Mat. Obsh. 23 (1970) (Russian) English translation: Trans. Moscow Math. Soc. 23 (1970) 243-285.