I first diagnosed the problem by replacing the burner control (Infinite switch) with one from a smaller burner (one that I knew worked fine). That seemed to solve the problem. I ordered the correct control for the big burner on Saturday night, it arrived Tuesday and installed it in less than 10 minutes using only a Phillips screwdriver.
Remember to unplug the stove before performing any service. Take this time to clean around and under the stove. (You know it needs it!)