Bring a large pot of water to a boil and add 2 tablespoons salt.
Heat the oil and butter in a large (12-inch) skillet over medium heat. When the butter stops foaming, add the onions and celery. Cook until softened, about 5 minutes
Stir in the mushrooms and 1 teaspoon salt. Cook another 5 minutes, or until the mushrooms have given up their liquid.
Add the garlic, porcini mushrooms and tomato paste and cook a few seconds until fragrant. Pour in the wine and let it bubble. Add the broth, tomatoes and thyme, stirring it all together. Bring to a simmer, then lower the heat to medium-low and cook 10-12 minutes, until the sauce is thickened.
While the sauce cooks, cook the pasta according to package directions, until al dente. Drain.
Transfer the pasta to the skillet, stirring to coat with the sauce. Taste for seasoning, and add more salt if needed. Sprinkle with the cheese and serve.
Notes
Use a mixture of mushrooms for an interesting texture - I used shiitake, cremini, and portobello. Plain button mushrooms are good too!
You can find dried porcini mushrooms in most supermarkets. They add a layer of flavor (umami!) but the sauce is equally delicious without them.