Soak the mushrooms in 1 cup hot water, 5 minutes. Strain through a paper towel-lined sieve, reserving the liquid. Rinse the mushrooms.
Bring a large pot of salted water to a boil. Add the tortellini and cook as the label directs. Reserve 1 cup cooking water, then drain.
Meanwhile, pulse the mushrooms, carrots, celery, onion and thyme in a food processor until finely chopped. Heat 4 tablespoons butter in a large Dutch oven or pot over medium-high heat. Add the vegetables, season with salt and pepper and cook, stirring, until they start browning, about 4 minutes. Add the wine and the reserved mushroom liquid; bring to a simmer, scraping up the browned bits. Cook until reduced by half, about 3 minutes.
Add the tomatoes to the pot and cook, stirring, until the sauce thickens, about 10 minutes. Add the basil, the remaining 2 tablespoons butter and the tortellini. Toss to coat, adding the reserved pasta water as needed to loosen.