Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 10 minutes. Drain, peel, and mash them in a bowl.
Combine mashed potatoes, 1 1/4 cups breadcrumbs, eggs, flour, Parmesan cheese, pecorino cheese, parsley, salt, and pepper in a bowl; mix by hand until smooth. Scoop up 2 to 3 tablespoons of the mixture; shape into an oval, cylindrical croquette. Repeat with remaining mixture.
Pour remaining 1 1/4 cups bread crumbs into a shallow plate. Dredge croquettes in the bread crumbs.
Heat olive oil in a large saucepan. Fry croquettes in the hot oil in batches until golden brown, about 2 minutes per side, turning as needed. Drain on paper towels.
Peel the hot potatoes by holding them with a fork, then peeling them with a knife.
Nutrition data for this recipe includes the full amount of breading ingredients. The actual amount of breading consumed will vary.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. The exact amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.