Melt the butter in an 8-inch nonstick skillet over medium-high heat. Add the corn, bell peppers, a generous pinch of salt, and a few grinds of pepper. Cook, stirring frequently, until tender, about 2 minutes, then reduce the heat to low.
In a small bowl, whisk the eggs. Add the eggs to the skillet and spread them and the corn mixture evenly around the pan. Cover and cook until the top is set, about 2 minutes. Lay the cheese on top of the egg and add the ham.
Turn off the heat and let sit, covered, until the cheese is melted, 30 seconds. Using a spatula, lift one edge of the omelet, fold it over by a third, and then continue to roll it out of the pan and onto a plate, seam side down. Serve immediately.