1 pound sliced cooked ham, chopped
1 pound domestic Swiss cheese, sliced and chopped
3 cloves garlic, minced
1/4 cup chopped flat-leaf parsley
1 teaspoon salt, plus more for cooking water
1 teaspoon black pepper
2 eggs, beaten
1/2 cup Parmesan cheese
1 package square wonton wrappers
1 egg beaten with 2 tablespoons water, reserved in small bowl