Preheat broiler. In extra-large skillet brown beef over medium-high heat; drain off fat. Stir in frozen vegetables. Cook, stirring occasionally, until heated through. Stir in half the cheese, half the parsley, 1/4 teaspoon salt, and 1/8 teaspoon pepper.
Meanwhile, in large bowl combine 2 cups boiling water, potato flakes, and 1 tablespoon of the butter. Stir until smooth. Season with salt and pepper; set aside.
Divide beef mixture among four 16-ounce broiler-safe dishes. Top with potatoes; sprinkle remaining cheese. Broil 3 inches from heat for 2 to 3 minutes, until cheese is melted. Drizzle with melted butter; sprinkle with remaining parsley. Makes 4 servings.