Adding cheese to polenta makes the texture even creamier. We used Fontina, which is a mild, nutty cheese that melts easily; Monterey Jack would also be good.
Place dried mushrooms in a medium bowl; add 4 cups warm water. Let soak until soft, about 30 minutes. Drain in a sieve lined with a paper towel or coffee filter, reserving liquid. Squeeze mushrooms to remove excess liquid; roughly chop.
In a 12-inch skillet, heat oil over medium heat. Cook onion, stirring often, until soft, about 5 minutes. Add garlic; cook until fragrant, about 30 seconds. Add fresh mushrooms; season with salt and pepper. Cover; cook until mushrooms release their juices, about 10 minutes. (If mushrooms begin to brown, add up to 1/4 cup water.)
Uncover; raise heat to high. Cook, stirring often, until liquid evaporates and mushrooms brown, 5 to 6 minutes. Add dried mushrooms, thyme, sherry, and tomato paste; cook until pan is dry, about 2 minutes. Stir in reserved liquid. Simmer, partially covered, until sauce has thickened, about 25 minutes. Stir in vinegar. Serve over soft polenta (see below).
To make soft polenta: In a small bowl, whisk together the yellow cornmeal with 1 cup cold water. In a medium saucepan, bring the remaining 3 1/4 cups water, coarse salt, and ground pepper to a fill rapid boil. Gradually stir in cornmeal mixture. Simmer, stirring frequently, until mixture is thick, 8 to 10 minutes. Remove from heat; stir in grated Fontina cheese and butter until melted.
Mixing the cornmeal with cold water before stirring it into the water in the pan prevents lumps from forming.