Heat oil in a large pot over medium-high heat. Add sausage and cook until golden brown, about 6 to 8 minutes. Transfer sausage to a plate lined with paper towels and reserve 2 tablespoons fat in the pot.
Reduce heat to medium. Add fennel, onion, salt, and pepper to the pot. Cook until vegetables are soft and beginning to brown, about 8 to 10 minutes. Stir in tomato paste and Italian seasoning, cooking until fragrant, about 1 minute.
Add pureed tomatoes, broth, water, Parmesan rind, and reserved sausage to the pot. Bring to a simmer, cover, and simmer until slightly thickened, about 15 minutes. Add noodles and simmer, uncovered, until al dente, about 12 to 14 minutes. Season with salt and pepper to taste.
In a small bowl, combine ricotta, grated Parmesan, salt, and pepper.
Ladle soup into bowls and top with cheese mixture, a drizzle of olive oil, and basil leaves. Enjoy your hearty Lasagna Soup!