Combine filling ingredients in 7 to 10-inch nonstick omelet pan or skillet. Cook and stir over medium heat until water has evaporated and vegetables are crisp-tender, 3 to 4 minutes. Remove from pan; keep warm. Clean pan.
Beat eggs, 1/4 cup water, cheese, basil and garlic powder in medium bowl until blended. Heat butter in same pan over medium-high heat until hot. Tilt pan to coat bottom. Pour in 1/2 of egg mixture. Mixture should set immediately at edges. Gently push cooked portions from edges toward center with inverted turner so that uncooked eggs can reach the hot pan surface. Continue cooking, tilting pan and gently moving cooked portions as needed. When top surface of eggs is thickened and no visible liquid egg remains, place 1/2 of filling on one side of the omelet. Fold omelet in half with turner and slide onto plate; keep warm. Repeat with remaining egg mixture and filling to make second omelet. Serve immediately. Serves 2