Spread 1 side of each bread slice with 1/2 teaspoon mustard. Divide turkey, Swiss cheese, and tomato slices evenly between 2 slices of coated bread. Sprinkle with pepper and top with remaining 2 bread slices, mustard side down.
Combine egg, egg white, milk, and chopped basil in a shallow dish; stir well with a whisk.
Coat a large nonstick skillet with cooking spray and place over medium heat until hot. Add 1/2 teaspoon butter. Dip both sides of each sandwich into egg mixture and place in pan. Cook 3 to 4 minutes on each side or until cheese melts and sandwiches are golden.
Cut sandwiches diagonally in half, and serve immediately.