Heat 1 tablespoon olive oil in a large pot over medium-high heat until shimmering. Add half the mushrooms in a single layer and cook undisturbed for 4-5 minutes until deeply browned on the bottom. Stir and cook 2 minutes more. Transfer to a plate. Repeat with another tablespoon oil and remaining mushrooms. Do not salt yet - salting too early prevents browning.

Reduce heat to medium. Add remaining 1 tablespoon oil, then onion with a pinch of salt. Cook 5-6 minutes, stirring occasionally, until softened and just starting to turn golden at the edges. Add garlic and cook 1 minute until fragrant.

Push onions to the side, add tomato paste to the cleared spot, and cook 2 minutes until it turns a deep rust color and smells sweet. Stir into onions, then add red wine, scraping up any browned bits from the bottom of the pot. Simmer 2 minutes until reduced by half.

Return mushrooms and any accumulated juices to the pot. Add crushed tomatoes, broth, and rosemary sprigs. Stir well, bring to a gentle simmer, then reduce heat to low. Cover partially and simmer 20-25 minutes, stirring occasionally, until thickened and the mushrooms are tender but still have some bite. The sauce should coat a spoon but not be pasty.

Remove rosemary stems. Off heat, swirl in cold butter and grated cheese until melted and glossy. Taste and adjust salt - you'll need more than you think. Stir in parsley. Serve over creamy polenta, pappardelle, or mashed potatoes, with extra cheese at the table.