Heat the olive oil in a large pot over medium heat. Brown the beef for 5 to 6 minutes, breaking it into small pieces. Drain excess fat if needed.
Add the chopped onion and cook for 3 to 4 minutes. Stir in the garlic, Italian seasoning, salt, pepper, and optional paprika; cook for 1 minute.
Stir in the tomato paste and cook for 1 to 2 minutes. Add the crushed tomatoes and beef broth.
Add the broken lasagna noodles. Bring to a gentle boil, then simmer for 12 to 15 minutes, stirring often, until the pasta is tender.
Taste and add the optional sugar if the tomatoes are too acidic. Adjust the seasoning.
In a small bowl, mix the ricotta, mozzarella, and Parmesan. Ladle the soup into bowls and add a spoonful of the cheese mixture. Garnish with parsley if desired.