Peel and cut onions into 1-centimeter rings, then separate the rings.
Slice the mozzarella into 4 even strips, and place a smaller onion ring in the center of a larger one. Fill the gap between the rings with strips of mozzarella. Repeat with remaining onion rings and freeze them for 1 hour.
Place the flour, eggs, and bread crumbs into three separate bowls. Dip each prepared ring into the flour, then the egg, then the breadcrumbs, then back into the egg, and one last time into the bread crumbs. Repeat with the remaining onion rings.
Heat the oil in a pot over high heat. Fry the rings until golden brown, being careful not to fry them for too long as the cheese may ooze out completely.