Creamy, starchy, salty, crunchy, cheesy -- here's plenty of proof that plain white can be every bit as thrilling to the taste buds as it is to the eyes.
Line a 9-inch-square baking dish with plastic, leaving overhang on 2 sides. Bring the water and coarse salt to a boil in a medium pan. Meanwhile, bring milk to a simmer in a small pan over medium heat.
Slowly add polenta to boiling water, whisking constantly until combined. Reduce heat until only a couple large bubbles appear at a time. Whisk in 1/4 cup simmering milk; cook, stirring often, until milk is absorbed, about 30 seconds. Repeat until all milk is added and polenta is cooked, about 5 minutes. Remove from heat; add grated Parmesan, stirring until melted.
Immediately pour polenta into prepared baking dish; smooth top. Let stand until no longer steaming, about 10 minutes. Refrigerate, uncovered, until cold and set, about 1 1/2 hours. Cover with plastic wrap, and refrigerate until ready to serve, up to 2 days.
Prepare mushrooms just before serving: Heat olive oil and butter in a medium skillet over medium-high heat until melted. Add mushrooms; cook, stirring frequently, until they have softened and released their liquid, about 5 minutes. Season with salt and pepper.
To serve, use plastic to lift polenta from baking dish; cut into 1-inch squares. Top with sauteed mushrooms and Parmesan cheese shavings.