Preheat oven to 375°F and cook lasagna noodles according to package directions, then drain and set aside.
In a large skillet over medium-high heat, brown the ground beef with diced onion and garlic until meat is cooked through, then drain excess fat.
Stir BBQ sauce, smoked paprika, salt, and pepper into the beef mixture and simmer for 5 minutes.
In a medium bowl, combine ricotta cheese, egg, and 1 cup of the cheddar cheese until well mixed.
Spread a thin layer of the BBQ beef mixture on the bottom of a 9x13-inch baking dish.
Layer 4 lasagna noodles over the sauce, followed by half of the ricotta mixture, one-third of remaining beef mixture, and one-third of the mozzarella.
Repeat layers once more, then top with remaining noodles, beef mixture, mozzarella, and remaining cheddar cheese.
Cover with foil and bake for 45 minutes, then remove foil and bake an additional 15 minutes until cheese is bubbly and golden.
Let stand for 10 minutes before slicing and serving.