Combine 3 tablespoons milk and egg whites in a shallow dish, stirring with a whisk. Combine 1/3 cup flour and next 4 ingredients (1/3 cup flour through pepper) in a shallow dish. Working with 1 steak at a time, dip in egg mixture; dredge in flour mixture. Repeat procedure with remaining steaks, egg mixture, and flour mixture.
Heat oil in a large nonstick skillet over medium-high heat. Add steaks; cook 3 minutes on each side or until browned. Remove steaks from pan; keep warm.
While steaks cook, prepare mashed potatoes according to package directions, using 1 1/3 cups milk. Keep warm.
Add mushrooms to pan; sauté 3 minutes. Combine 2 1/2 tablespoons flour, 1/4 teaspoon salt, and broth, stirring with a whisk. Add broth mixture to pan. Bring to a boil; cook 1 minute, stirring constantly. Spoon over steaks. Serve with mashed potatoes.