Marinate the chicken: Arrange the cubed chicken in a shallow medium bowl and toss with the marinade ingredients, using 1/4 cup of olive oil. Season lightly with salt and let marinate for about 1 hour, refrigerated. 



In a skillet or sauté pan large enough to hold the chicken in one layer, heat the remaining 1 tablespoon of oil. Remove the chicken from the marinade with a slotted spoon. Sauté the chicken just to brown on all sides. Do not overcook. Remove from the skillet with a slotted spoon and set aside. 



Place a pizza stone on the middle rack of the oven and preheat the oven to 500 degrees F. On a lightly floured surface, stretch or roll out the dough to make two 10-inch rounds, with the outer border a little thicker than the inner circle. Transfer dough to wooden pizza paddles or cookie sheets. 



Layer the pizza: Start with the mozzarella, then the Fontina, tomato, eggplant, caramelized onions, cooked chicken, and peppers. Slide the pizza onto the baking stone. Bake until the pizza is nicely browned, 10 to 12 minutes. 



Transfer the pizza to a firm surface and cut into slices with a pizza cutter or very sharp knife. Serve immediately.