Melt garlic butter in 8 or 9-inch skillet until sizzling; add broccoli, carrot and green onions. Cook over medium heat, stirring occasionally, until crisply tender (4 to 5 minutes).
Stir together eggs, water and salt in medium bowl; add to vegetable mixture. Sprinkle with basil. Cover and cook over medium heat until center is almost set (4 to 6 minutes).
Sprinkle with cheese. Cover; remove from heat. Let stand until cheese is melted (2 to 3 minutes).
To serve, cut into wedges. Garnish with sliced chiles, if desired.