In a bowl, beat egg whites and oil to blend. In another bowl, mix crumbs, basil, and 2 teaspoons garlic.
Add 1 kind of vegetable at a time to the egg mixture—eggplant, zucchini, bell pepper, and 1 1/2 cups tomatoes. Lift out with a slotted spoon, draining briefly, and add to crumbs. Mix gently to coat, the lift out. Arrange vegetables in a single layer on a 14-inch by 17-inch baking sheet.
Bake in a 450° oven until vegetables are well-browned and eggplant is soft when pressed, 20 to 25 minutes.
Meanwhile, in a blender or food processor, purée until smooth the remaining 1 teaspoon garllic, remaining 1/2 cup tomatoes, mayonnaise, and lemon juice to taste.
Spoon vegetables onto warm dinner plates and season to taste with mayonnaise mixture and salt and pepper.