Creamy Polenta with Mushroom Ragù

Name: Creamy Polenta with Mushroom Ragù
Ingredients: 2 pounds mixed mushrooms, such as cremini, shiitake, and oyster; 1 medium yellow onion; 1 medium carrot; 3 cloves garlic; 2 tablespoons olive oil; 1 teaspoon kosher salt, divided; 1/4 teaspoon freshly ground black pepper; 1/4 cup dry white wine; 2 tablespoons tomato paste; 1 (28-ounce) can diced tomatoes; 1/2 cup water; 1 bay leaf; 2 sprigs fresh thyme; 1 sprig fresh rosemary; 1/4 teaspoon red pepper flakes; 1 teaspoon balsamic vinegar; 4 cups water; 1 cup polenta or yellow cornmeal; 1 teaspoon kosher salt; 1/2 cup grated Parmesan cheese, plus more for serving; 2 tablespoons unsalted butter