Soak the porcini in 1 cup hot water for 30 minutes. Drain, straining and reserving the soaking liquid. Chop the porcini and set aside.
Heat the oil in an 11- to 12-inch straight-sided sauté pan over medium heat. Add the pancetta, celery, carrot, onion, and 1 tsp. salt; cook, stirring often, until the vegetables are soft and lightly golden, about 15 minutes. Add the garlic and cook, stirring, for 1 minute. Add the brisket and its juices, wine, thyme, and the porcini and their soaking liquid. Bring to a simmer, reduce the heat to low, and cook until the sauce is very thick, about 30 minutes. Season to taste with salt and pepper.
Meanwhile, bring a large pot of well-salted water to a boil. Cook the pasta in the boiling water until al dente. Drain and toss with most of the meat sauce. Top with the remaining sauce and pass the Parmigiano at the table.