Preheat oven to 475° with a rack in top third of oven and another in lower third. Combine mushrooms, oil, garlic, thyme, pepper, and 1/2 tsp. salt in a large bowl; toss to coat. Divide seasoned mushrooms between two large rimmed baking sheets and spread in an even layer.
Bake mushrooms 10 minutes, then remove from oven and increase heat to broil. Broil mushrooms, one baking sheet at a time, until browned, 5 minutes each.
Meanwhile, cook pasta according to package instructions. Drain, reserving at least 1/2 cup pasta water.
Transfer broiled mushrooms to a bowl. Scrape browned bits from one baking sheet onto second baking sheet. Set baking sheet with browned bits on stovetop over medium-high heat. Add broth and cook, stirring and scraping to loosen browned bits from bottom of baking sheet, until mixture boils and reduces slightly, 1 to 2 minutes.
Combine mushroom mixture, pasta, half of parsley, half of tarragon, and remaining 1/2 tsp. salt in a large bowl; toss to coat. Divide among four shallow bowls, then drizzle with broth mixture and a bit of pasta water if needed. Top with remaining parsley, tarragon, and the parmesan.
Wine pairing: FEL 2014 Pino Noir (Anderson Valley; $38)