Heat the olive oil in a large Dutch oven or heavy soup pot over medium heat. Add the chopped onion and grated carrot and cook, stirring occasionally, until softened and lightly golden, about 6–7 minutes.
Stir in the minced garlic and cook 30–60 seconds until fragrant. Push the vegetables to the side, add the sausage and break it up with a spoon. Brown the meat, stirring occasionally, until no pink remains — about 7–9 minutes. Drain excess fat if there is more than 1–2 tablespoons.
Mix the Italian seasoning, black pepper and red pepper flakes into the meat and vegetables. Add the tomato paste and cook, stirring frequently, for 1–2 minutes to deepen the flavor.
Pour in 2 cups of the pasta sauce and use it to deglaze the pot, scraping up any browned bits. Add the diced tomatoes, the remaining pasta sauce and the broth, then tuck in the bay leaf. Bring the mixture up to a gentle boil.
Once boiling, stir in the broken lasagna noodles. Reduce the heat to a simmer and cook, uncovered, stirring occasionally so noodles don’t stick, until the pasta is tender but still holds its shape, about 12–15 minutes (timing depends on noodle size).
Meanwhile, combine the cottage cheese, balsamic vinegar and nutmeg in a blender or food processor and puree until smooth and slightly airy. Set aside.
When the noodles are just cooked, remove and discard the bay leaf. Stir the spinach into the soup and then fold in the blended cottage-cheese mixture to create a creamy, lasagna-like texture. Simmer 2–3 minutes more so the flavors come together and the spinach wilts.
Taste and adjust seasoning with salt and additional black pepper if needed. Ladle the soup into bowls and top each serving with shredded mozzarella, a sprinkle of Parmesan and torn basil leaves. Serve immediately.