Two-stage stochastic programs are a class of stochastic problems where uncertainty is discretized into scenarios, making them amenable to solution approaches such as Benders decomposition. However, classic Benders decomposition is not applicable to general two-stage stochastic mixed-integer programs due to the restriction that the second-stage variables should be continuous. We propose a novel Benders decomposition-based framework that accommodates mixed-integer variables in both stages as well as uncertainty in all of the recourse parameters. The proposed approach is a unified branch-and-Benders algorithm, where we use a heuristic to maintain a global upper bound and a post-processing phase to determine an optimal solution. This new approach is flexible, allowing practitioners to integrate acceleration techniques such as partial decomposition or convexification schemes. We demonstrate the efficiency of our approach versus classic ones on the stochastic server location problem; and, its generality on a new, complex stochastic problem where the second stage is a traveling salesman problem.