Heat oil in a large pot over medium-high heat. Add onion and cook, stirring occasionally, until almost translucent, about 5 minutes. Add sausage and cook, breaking up the sausage into small pieces with a wooden spoon, until browned, about 4 minutes. Add garlic and tomato paste; cook, stirring constantly, until the tomato paste is heated through, about 2 minutes. Add tomatoes, water, broth, sugar and salt; bring to a boil over high heat. Add lasagna noodles and stir to separate. Cook, stirring occasionally, until the pasta is just cooked through but not completely soft, 8 to 10 minutes. Remove from heat and stir in vinegar.

Meanwhile, combine ricotta, mozzarella and Parmesan in a small bowl; set aside.

Ladle the soup into 6 bowls. Top each serving with a dollop of the ricotta mixture. Top with fresh basil.