For the polenta, put 4 cups of cold water into a medium, heavy-bottomed pot with the 1 cup of milk, if using. Alternatively use 5 cups cold water. Stir in the polenta and the teaspoon of kosher salt. Bring to a boil over medium-high high, stirring often. Reduce the heat to medium-low and cook the polenta stirring occasionally, until it is tender, 45 to 60 minutes. The polenta will swell and thicken as it cooks. If it gets too thick before it's finished cooking, stir in more water as needed. Stir in the butter. Taste, and add more salt as needed.
Meanwhile, make the roasted mushrooms: Preheat the oven to 325° F. Remove the stems from the shiitakes (and reserve for stock-making or the compost pile or discard). Trim off the very bottom end of the cremini stems and discard. Quarter the mushrooms.
In a 9- by 13-inch baking dish, combine the mushrooms, thyme sprigs, shallots, garlic, sherry, and olive oil. Season with a big pinch of salt and pepper to taste. Toss to distribute everything evenly, then spread in a an even layer. Cover pan with foil, transfer pan to the oven, and cook for 30 minutes.
Remove pan from the oven and raise the heat to 450° F. Remove the foil. Holding the mushrooms in the pan with a spatula, drain the juices into a small saucepan and set aside. Spread the mushrooms out again, return pan to the oven, and roast until the mushrooms are beginning to brown, 15 to 20 minutes longer, stirring after 10 minutes.
Bring the juices to a gentle simmer. Let the juices reduce slowly while the mushrooms finish cooking—they don't need to reduce by half or any particular amount, but the flavors will concentrate as it simmers/reduces. When the mushroom are nearly done, add the tablespoon of butter to the juices. When it has incorporated into the juices, taste them. Adjust seasoning as needed with more salt or butter to taste.
To serve: Ladle polenta into bowls, spoon mushrooms over top, drizzle with sauce.