Heat 2 tablespoons of oil in a heavy-bottomed saucepan over medium heat. Once hot, add onions and sauté until tender, about 5 minutes. Add sliced mushrooms and sauté until cooked through. Transfer to a bowl and set aside.

In the same saucepan, heat an additional tablespoon of oil over medium heat. Add greens and cook until wilted and tender. Transfer to bowl with mushrooms and set aside.

In the same saucepan, heat an additional tablespoon of oil over medium heat. Add garlic and cook until fragrant, but not browned, about 2 minutes. Add tomatoes, reduce heat, and simmer 10-15 minutes.

Break apart tomatoes, add basil, taste and add salt and black pepper, if needed. Remove from heat.

Spread a bit of sauce on the bottom of the pan. Roll out pasta as thin as possible and place a layer of pasta on the sauce. Add about 1/3 of the Swiss Chard and mushrooms, 1/4 of the cheese, and about 1/4 of the sauce.

Continue layering Swiss Chard, mushrooms, sauce, and cheese until you have three layers. Top with a layer of sauce and pasta, then cover with cheese.

Cover with baking parchment, and bake for 45 minutes. Remove parchment and bake another 15 minutes or until bubbly and browned.

Remove from the oven and let settle for 5 to 10 minutes, then serve.