1. In medium saucepan, bring the water and 1/3 cup of milk to boil.
2. Remove the pan from the heat and stir in the instant mashed potatoes. Allow the potatoes to rest for 5 minutes (the potatoes will be stiff)
3. Stir in 1/4 cup of the Parmesan cheese, the salt, and the pepper.
Cover the potato dough with plastic wrap and refrigerate until cold.
4. Using a tablespoon, scoop the potatoes an roll between your palms to form 1 1/2 inch (3.8cm) balls.
Push a cheddar cube into the center of each potato ball and roll again to seal.
5. Place the flour in a shallow bowl. In a separate bowl, beat together the egg and the remaining teaspoon of milk.
6. Combine the bread crumbs and the remaining tablespoon of Parmesan cheese in a shallow dish.
7. Roll the potato balls in the flour, dip in the egg mixture, and then, dredge in the bread crumbs.
8. Fill a saucepan halfway with vegetable oil and heat over medium-high heat until a deep-fat thermometer reads 360F(182C).
9. Fry the potato croquettes for 2 to 3 minutes, until golden brown. Drain on paper towels and serve warm.