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Preparation Place racks in upper and lower thirds of oven; preheat to 325°F. Combine mushrooms, thyme sprigs, and garlic on a large rimmed baking sheet. Season generously with kosher salt and pepper; drizzle with oil. Toss to coat mushrooms, then spread out in an even layer. Transfer to upper rack in oven and let mushrooms roast while you prepare polenta.
Bring 4 1/2 cups water to a simmer in a large ovenproof saucepan over medium-high heat. Add butter and a generous pinch of kosher salt and whisk to melt butter. Gradually add polenta, whisking constantly. Return mixture to a boil, immediately cover pot, and transfer to lower rack in oven. Bake polenta, shaking baking sheet with mushrooms occasionally, until polenta is tender, 25–30 minutes.
Remove polenta from oven. Crank up oven temperature as high as it will go (but don’t broil). Continue to cook mushrooms until crisp around the edges, 5–10 minutes longer.
Meanwhile, carefully uncover polenta and whisk vigorously, scraping bottom of pan, until polenta is smooth and thick. Gradually add 4 oz. Parmesan, whisking constantly until melted and incorporated; taste and season with more kosher salt and pepper. Cover and keep warm over low heat while mushrooms finish roasting.
Remove mushrooms from oven; drizzle with vinegar. Toss to coat; let cool slightly.
Divide polenta among bowls and top with mushrooms, thyme leaves, sea salt, and more Parmesan.