Heat a large deep frying pan (not nonstick) over medium-high heat until a drop of water skitters across its surface. Swirl in oil, add mushrooms, and cook undisturbed 4 minutes. Reduce heat to medium and stir in shallots. Cook, stirring occasionally, until mushrooms have released all their liquid and shallots are tender, 7 to 8 minutes. Stir in tomato paste and thyme; cook 1 minute. Stir in wine, scraping up any bits from bottom of pan, and cook until mostly evaporated, 3 to 5 minutes.
Add broth, 1/2 tsp. salt, and the pepper. Simmer until sauce thickens and mushrooms are tender, 10 to 12 minutes. Remove from heat and stir in butter and 2 tbsp. parsley.
Meanwhile, bring 1 qt. water and remaining 1/2 tsp. salt to a boil in a medium saucepan. Stirring constantly, add polenta. Cook, stirring, until polenta thickens, about 3 minutes. Remove from heat and stir in 1 3/4 cups cheese. Cover and set aside until ready to serve.
Just before serving, stir polenta and divide among four deep, wide bowls. Top with mushroom ragout, remaining 1/4 cup cheese, and remaining 2 tbsp. parsley.