Heat the oil in a Dutch oven over medium-high heat. When the oil is hot, add the mushrooms, thyme and salt, then stir. Cook until the liquid is released and the mushrooms are browned, 6 to 8 minutes.
Add the Italian seasoning and garlic, stir and cook for 1 minute. Sprinkle with the flour, stirring until the flour has coated all the mushrooms. Stir in the broth and bring to a simmer; simmer, stirring occasionally, until the mixture thickens, about 10 minutes.
Add the heavy cream and return to a simmer. Add the lasagna sheets and cook until al dente, 3 to 4 minutes. Add the spinach and cook just until the spinach wilts, about 1 minute.
Turn the heat off and ladle into bowls. Garnish with a few dollops of ricotta cheese and a few drizzles of pesto.