Preheat oven to 350 degrees F. Grease a 3-quart rectangular baking dish; set aside. In 12-inch skillet cook zucchini, sweet pepper, onion, eggplant, garlic, 1/2 teaspoon salt, and 1/4 teaspoon ground black pepper in hot oil over medium-high heat for 10 to 15 minutes or until vegetables are tender, stirring occasionally.
Meanwhile, in extra-large bowl whisk together eggs and mayonnaise until combined. Stir in Pecorino-Romano cheese and 1 cup of the mozzarella cheese. Add cooked vegetables; toss to combine. Evenly spread vegetable mixture in prepared baking dish. Top with remaining mozzarella and cracker crumbs.
Bake, uncovered, for 20 to 25 minutes or until top is lightly browned and a knife inserted near center comes out clean. Let stand 10 minutes before serving. Makes 6 to 8 servings.