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, 3-4 minutes. Remove from pan; keep warm. Clean pan.
Beat eggs, 1/4 cup water, cheese, basil, 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 half of the egg mixture. Mixture should set immediately at edges. Gently push cooked portions from edges toward the center with inverted spatula and slide onto plate; keep warm. Repeat with remaining egg mixture and filling to make second omelet. Serve immediately.
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