Unwrap mozzarella sticks. If adding cheddar, press shredded or thin strips along the mozzarella. Freeze for 30–40 minutes so they hold up during frying.
Set up three bowls: flour with a pinch of salt and pepper; whisked eggs with milk; and a mix of seasoned breadcrumbs, panko, garlic powder, onion powder, paprika, Parmesan, salt, and pepper.
Roll frozen sticks in flour, dip in egg wash, then coat with breadcrumbs. For extra crunch, repeat the egg wash and breadcrumbs. Arrange on a parchment-lined tray and freeze again for 30 minutes.
Heat oil to 350°F (175°C) and fry 3–4 sticks at a time for 2–3 minutes until crisp and golden. Drain on paper towels and serve hot.
For a lighter option, preheat the air fryer to 390°F (200°C). Lightly spray sticks with cooking spray and air fry 6–8 minutes, flipping halfway, until golden and crispy.