Heat 2 tablespoons of olive oil in your pot over medium heat. Add the finely diced onion, carrot, and celery. Cook for 6-7 minutes, stirring occasionally, until soft but not browned.
Add the finely chopped mushrooms and a generous pinch of salt. Cook for 8-10 minutes, stirring occasionally, until deeply browned.
Add the tomato paste and grated garlic. Cook, stirring constantly, for 2 minutes until the paste darkens and smells deeply sweet.
Pour in the crushed tomatoes and 4 cups of vegetable broth. Add the Parmesan rind if using. Bring to a boil, then reduce to a simmer. Let it run for 15 minutes. Taste and adjust salt.
Add the broken lasagna sheets. Simmer for 8-10 minutes, stirring gently twice, until al dente. Pull the pot off the heat immediately.
Stir in half a cup of ricotta and a generous handful of grated Parmesan. Taste for acid and salt. Add a squeeze of lemon if needed.
Ladle into wide bowls. Drop a generous spoonful of the remaining ricotta in the center. Use the back of the spoon to swoosh it out. Drizzle with olive oil. Place two or three basil leaves in the center.