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Drain the cooked potatoes. When they are cool enough to handle, run the cooked potatoes though a food mill. Add 2 tablespoons olive oil and the 2 eggs. Mix with a wooden spoon until well combined. Add both cheeses, and season, to taste, with salt and pepper. Using the wooden spoon, mix again until well combined.
In a separate bowl, or the bowl of an electric stand mixer fitted with a whisk attachment, whisk the egg white until it forms glossy firm peaks. Using a spatula, gently fold the beaten egg white into the potato mixture. The goal here is to add volume to the potato mixture, not to deflate the egg whites. Gently fold in the bread crumbs.
Fill a deep pot no more than halfway with extra-virgin olive oil. Heat oil over medium-high heat until it reaches a temperature of 350 F degrees. The oil should remain at or around this temperature throughout the cooking process.
Using a tablespoon, measure out a full tablespoon of the potato mixture to form the crocche. You can use your hand to gently mold the mixture into more of a circle.
Working in batches of 6, fry the potato crocche until golden brown. Use a slotted spoon or spider to gently drop the fritters into the oil, being careful not to splatter the hot oil. Using the same spoon or spider, remove the finished fritters to a serving plate. Repeat process until all the crocche are cooked.