Combine eggs, cheeses, basil, 1/4 teaspoon salt, and 1/4 teaspoon black pepper in a large bowl; stir well. Set aside.
Heat oil in a 12-inch nonstick skillet over medium-high heat until hot. Add onion, garlic, 1/4 teaspoon salt, and 1/4 teaspoon black pepper; sauté until tender. Add bell pepper and bay leaf to pan; sauté until very tender. Discard bay leaf. Spread bell pepper mixture evenly in bottom of pan. Reduce heat to medium-low. Stir egg mixture, and pour over bell pepper mixture in pan. Cook 3 to 5 minutes or until edges begin to set. Wrap handle of pan with foil. Transfer pan to oven, and bake, uncovered, at 375° for 6 to 8 minutes or until eggs are completely cooked.
Loosen sides and bottom of frittata with a spatula. Turn out onto a plate. Brush bottom and sides with balsamic vinegar. Cut into 6 wedges, and serve immediately.