This paper tackles the optimal traffic signal control of isolated oversaturated intersections. An analytical signal control algorithm is proposed to find the global optimal signal timings with dynamic cycle lengths and phase splits to minimize the vehicle delay throughout the oversaturation period at a generic multi-phase junction. The traffic dynamics are modelled based on the kinematic wave theory and the predicted traffic flows. Moreover, spillback avoidance is incorporated during the queue formation oversaturated regime by adopting a mixed delay and probability of spillback objective function. Microsimulation experiments demonstrate the optimality, practicality, and robustness to system uncertainties of the proposed signal control method. The results pinpoint over 63%, 55%, and 40% reduction in total vehicle delay by implementing the proposed signal control respectively compared to an optimal fixed-time, actuated, and capacity-aware max pressure signal control methods.