Boil the potatoes until fork tender, and let cool for 20 minutes.
In a large bowl, mash the potatoes. (For a creamier texture, use a potato ricer.) Add the Parmigiano cheese, chopped parsley, and season with salt and pepper. Add one egg and mix well, until all the ingredients are evenly distributed.
Grab a handful of mashed potatoes and press into logs that are 1 1/2-inches thick by 3 inches long. With your finger, create an indentation in the middle of the log and add a slice of mozzarella. Close up the logs and make sure the mozzarella is fully covered. Reshape if necessary.
For the batter, dip potato logs in a beaten egg and cover with breadcrumbs.
Heat the extra-virgin olive oil in a large pot to 350 degrees F.
Fry in the oil in batches until the potato croquettes are crispy and golden brown.
Let drain on a paper towel lined plate, and serve warm.