Preheat the oven to 450 degrees F.
Bake the potatoes right on an oven rack until a bit overcooked, about 1 hour. Let sit until cool enough to handle, cut in half, and scoop out the flesh. Reserve the potato skins, if desired, for another use.
Lower the oven to 250 degrees F.
Pass the potatoes through a potato ricer or grate them on the large holes of a box grater. You should have about 6 cups. Season the potatoes with the salt and pepper, and then divide them into 16 equal balls. Press each ball into a patty about 1/3-inch thick.
Cut half of the cheese into 1/4-inch pieces. Place a piece of cheese in the middle of each of 8 patties. Cover with the remaining patties. Shape the patties into smooth disks. Lightly dust the patties on both sides with flour. (Potato patties can be made ahead and covered and refrigerated up to 1 day before cooking.)
Heat a 12-inch skillet over medium-high heat. Add 3 tablespoons of oil and heat until the oil just begins to smoke. Place 4 patties in the skillet. When you hear the patties sizzle, lower the heat to medium. Cook until brown and crispy on both sides, 10 to 12 minutes total, turning once. If the potatoes absorb all the oil, add another tablespoon or 2 of oil to the skillet and cook the remaining 4 patties.
In a microwave-safe bowl, warm the pesto in the microwave for about 30 seconds or until warm throughout. Grate the remaining mozzarella.
To serve: Sprinkle each potato cake with grated mozzarella and place in a 250 degrees F oven until the cheese is melted. Spoon pesto on top of each cake and serve immediately.