The Connected Max Cut (CMC) problem takes in an undirected graph G(V, E) and finds a subset S ⊆ V such that the induced subgraph G[S] is connected and the number of edges connecting vertices in S to vertices in V \ S is maximized. This problem is closely related to the Max Leaf Degree (MLD) problem. The input to the MLD problem is an undirected graph G(V, E) and the goal is to find a subtree of G that maximizes the degree (in G) of its leaves. [Gandhi et al. 2018] observed that an α-approximation for the MLD problem induces an O(α)-approximation for the CMC problem. We present an O(log log |V |)-approximation algorithm for the MLD problem via local search. This implies an O(log log |V |)-approximation algorithm for the CMC problem. Thus, improving (exponentially) the best known O(log |V |) approximation of the Connected Max Cut problem [Hajiaghayi et al. 2015].