Bring 4-1/2 cups water to a boil in a 3-quart saucepan. Whisk in 1-1/2 tsp. salt and then the cornmeal. Reduce the heat to medium low and cook, stirring occasionally, until the cornmeal is tender and absorbs all the liquid, about 30 minutes.
Meanwhile, make the mushroom ragoût: Heat the oil in a medium (10-inch) skillet over medium-high heat. Add the fresh mushrooms and cook, stirring, until they soften,3 to 4 minutes. Reduce the heat to medium, add the garlic, and cook, stirring, until fragrant, about 1 minute. Add the rehydrated mushrooms and their liquid, thyme, 1/4 tsp. salt, and 1/2 tsp. pepper; cover with the lid partially ajar and cook, stirring occasionally, until the liquid is a saucy consistency and the mushrooms soften, 7 to 8 minutes.
Stir the Fontina into the polenta. Taste and season with salt as needed. Serve the polenta topped with the ragoût.