Place each piece of chicken between 2 sheets of wax paper or in food storage bag. Pound gently to flatten, using a rolling pin or smooth side of meat mallet. Sprinkle chicken pieces with salt and pepper. Combine eggs with 1 cup milk; dip chicken pieces in mixture. Place a piece of ham and a piece of cheese in the center of each flattened piece of chicken. Brush top of cheese slices with the milk and egg mixture. Fold ends over the ham and cheese; roll up, beginning with long side. Secure rolls with toothpicks. Dredge chicken in the 1/3 cup flour then dip in remaining milk mixture; roll in bread crumbs to coat well. Place chicken rolls on a plate; cover and refrigerate for 1 hour.

Heat 1/2-inch of vegetable oil in a heavy skillet over medium-low heat. Fry chicken rolls for 20 minutes, turning frequently, until golden brown. Drain well. Serve with mushroom sauce. Serves 8.

Mushroom Sauce

Melt butter in a saucepan over medium heat; sauté mushrooms until browned. Add flour and stir until mixture is well blended. Gradually stir in hot milk and chicken broth. Cook over medium heat, stirring constantly, until sauce begins to boil and thickens. Simmer, stirring frequently, over very low heat for 5 minutes. Season with salt and pepper to taste. Makes about 1 cup of medium thick sauce.