1. Bring 4 1/2 cups water to a boil in a large, heavy-bottomed pot; season with salt. Bring another 4 cups water to a simmer in a small saucepan. Gradually add polenta to large pot, whisking constantly until combined. Reduce heat until only one or two large bubbles break surface at a time, adjusting heat as necessary.

2. Whisk 2 ladlefuls of simmering water into polenta and cook, stirring frequently with a wooden spoon, until water has been absorbed, about 5 minutes. Continue to add 2 ladlefuls of simmering water every 5 minutes, stirring often and waiting for it to be absorbed before adding more, until polenta is creamy and just pulls away from sides of pot, about 45 minutes. (Adjust heat as necessary.)

3. Pour polenta into a slightly damp 9-by-13-inch baking dish. Let stand until no longer steaming, about 10 minutes. Refrigerate, uncovered, until cold and set, about 1 1/2 hours.

4. Preheat oven to 450 degrees. In a bowl, toss together mushrooms, shallot, thyme, and oil; season with salt.

5. Turn out polenta onto a cutting board. Cut into 24 pieces, and transfer to a rimmed baking sheet brushed with oil. Dividing evenly, brush tops with oil, then sprinkle with cheese and top with mushroom mixture. Roast until polenta is crisp on bottom, cheese is melted, and mushrooms are wilted and golden brown, about 25 minutes. Serve immediately.