Preheat the oven to 375 degrees F. Lightly grease a 9x9 inch baking dish.
Melt 2 tablespoons of the butter in a large skillet over medium heat. Add the sliced leeks and cook until softened, about 5 minutes.
Add the remaining tablespoon of butter along with the mushrooms and garlic. Cook until the mushrooms release their liquid and begin to brown, about 8 minutes.
Stir in the heavy cream, thyme, salt, and pepper. Simmer for 3 minutes until slightly thickened, then remove from heat.
Spread a thin layer of the mushroom mixture in the bottom of the baking dish. Top with 4 noodles, half the ricotta, and one third of the mozzarella.
Repeat with another layer of mushroom mixture, the remaining noodles, the rest of the ricotta, and half the remaining mozzarella.
Finish with the last of the mushroom mixture and the remaining mozzarella and Parmesan.
Cover the dish tightly with foil and bake for 25 minutes. Remove the foil and bake 10 more minutes until the top is golden.