Bring a large pot of lightly salted water to a boil, and stir in the pasta. Cook 8 to 10 minutes, until al dente, and drain.
Heat 3 tablespoons olive oil in a skillet over medium heat. Whisk together the egg and water in a bowl. Place bread crumbs in a separate bowl. Dip chicken pieces first into the egg mixture, then into the bread crumbs to coat. Place coated chicken pieces in the skillet, and cook 5 minutes on each side, or until coating is golden brown and juices run clear. Drain on paper towels.
Heat remaining 1 tablespoon olive oil in a separate skillet over medium heat. Mix in garlic, onion, green bell pepper, red bell pepper, yellow bell pepper, and mushrooms. Cook and stir until vegetables are tender.
In a large bowl, toss the pasta, chicken, and vegetables with the Greek dressing. Serve topped with tomatoes and Parmesan cheese.
