In a medium bowl, combine the shredded mozzarella, shredded cheddar, 3 tablespoons of flour, 1 beaten egg, dried oregano, and black pepper. Mix well until the ingredients bind together into a dough-like consistency.
Scoop about 2 tablespoons of the cheese mixture and shape it into a ball, then flatten it slightly. Use your finger to poke a hole in the center to form a ring shape. Repeat with the remaining mixture.
Place the shaped rings on a parchment-lined baking sheet and freeze for at least 30 minutes. This step is crucial to prevent the cheese from melting too fast during frying.
Prepare a breading station with three shallow bowls: one with the remaining 1 cup of flour, one with the 2 beaten eggs, and one with the Panko breadcrumbs.
Remove the rings from the freezer. Dredge each ring in the flour (shake off excess), dip into the beaten egg, and then coat thoroughly with Panko breadcrumbs.
Heat the vegetable oil in a deep pan or pot to 350°F (175°C).
Fry the rings in batches for 2-3 minutes, or until golden brown and crispy. Be careful not to overcrowd the pan.
Transfer the fried rings to a wire rack or paper towel-lined plate to drain excess oil. Serve immediately with marinara sauce.