Boil the Potatoes: Place the potatoes in a large pot and cover with cold water. Bring to a boil and cook until tender, about 15-20 minutes. Drain well.

Rice or Mash: While still hot, rice the potatoes or mash them thoroughly. Ricing will yield a smoother texture, but mashing works just fine. Be sure to eliminate lumps.

Combine Ingredients: In a large mixing bowl, combine the mashed potatoes, egg, flour, Parmesan cheese, chives, salt, and pepper. Mix well until all ingredients are thoroughly incorporated.

Form the Cakes: Take a generous tablespoon of the potato mixture and flatten it into a disc in the palm of your hand. Place a cube of mozzarella cheese and a cube of cheddar cheese in the center. Gently bring the edges of the potato mixture up and around the cheese, forming a ball. Flatten the ball slightly into a cake shape. Repeat with the remaining potato mixture and cheese.

Dust with Flour: Lightly dust each potato cake with flour, ensuring that they are evenly coated. This will help them crisp up nicely in the pan.

Fry the Cakes: Heat the vegetable oil in a large skillet over medium heat. Once the oil is hot, carefully place the potato cakes in the skillet, being careful not to overcrowd the pan.

Cook to Golden Perfection: Cook for about 3-4 minutes per side, or until golden brown and crispy. The cheese should be melted and gooey inside.

Serve Immediately: Remove the potato cakes from the skillet and place them on a paper towel-lined plate to drain any excess oil. Serve immediately with sour cream or your favorite dipping sauce.