In a 2- to 2 1/2-quart pan, combine rice and 2 3/4 cups water. Bring to a boil over high heat and cook until most of the water is absorbed, 7 to 10 minutes. Reduce heat to low, cover, and simmer until rice is tender to bite, 10 to 15 minutes longer. Scrape rice into a large bowl. Cool about 5 minutes, stirring occasionally with a fork.
In another bowl, beat eggs to blend. Add eggs and cheddar cheese to rice and mix well. Scrape rice mixture into an oiled 9- by 13-inch pan and pat into an even layer.
Bake in a 400° oven until rice begins to brown around edge, 15 to 20 minutes.
Meanwhile, in a 10- to 12-inch frying pan over high heat, frequently stir 1 tablespoon olive oil, bell pepper, mushrooms, zucchini, and onion until mushrooms begin to brown, 3 to 5 minutes. Stir in olives. Remove from heat.
Spread pizza sauce evenly over baked rice and sprinkle with mozzarella cheese. Spoon vegetable mixture over crust and top with parmesan cheese.
Return to oven and bake until parmesan begins to brown, 8 to 10 minutes. Add salt and pepper to taste. Cut into rectangles and lift from pan with a spatula.