01 - Separate onion slices into individual rings. Set up three shallow bowls: mix flour with smoked paprika, garlic powder, and salt in the first; whisk eggs in the second; place panko breadcrumbs in the third.
02 - Dredge each onion ring in the seasoned flour mixture, shaking off excess. Dip into beaten eggs, then press firmly into panko breadcrumbs to coat evenly. Transfer to a baking sheet or plate.
03 - Pour vegetable oil into a deep skillet to reach 2 inches depth. Heat over medium-high heat until oil reaches 350°F. Fry onion rings in batches for 2-3 minutes until golden brown and crispy. Remove with tongs and drain on paper towels.
04 - Heat a non-stick skillet over medium heat. Place one tortilla in the pan. Sprinkle evenly with 1/2 cup cheddar and 1/4 cup mozzarella cheese. Arrange fried onion rings over the cheese, then drizzle with 2 tablespoons sour cream and scatter with chives if using.
05 - Top with second tortilla. Cook for 2-3 minutes until bottom tortilla is golden brown. Carefully flip using a large spatula. Cook another 2 minutes until cheese is completely melted and second side is golden. Transfer to cutting board. Repeat with remaining ingredients.
06 - Let quesadillas rest for 2 minutes. Cut each into 4-6 wedges. Serve immediately with salsa, guacamole, and additional sour cream on the side.