Heat 2 tablespoons of the olive oil in a Dutch oven or large, deep skillet over medium heat. Add the mushrooms, ½ teaspoon of the salt, and several grinds of pepper and toss to coat. Cook, stirring only occasionally, for 8 minutes, or until soft and browned. Remove from the pan and set aside. Work in batches if necessary.

Wipe out the pan and return it to the heat. Add the remaining 2 tablespoons olive oil, the onion, and the remaining ¼ teaspoon sea salt. Cook, stirring occasionally, for 5 to 8 minutes, or until softened. Stir in the garlic, thyme, and rice. Let cook for 1 minute, then add the wine. Stir and cook for 1 to 3 minutes, or until the wine cooks down.

Add the broth ¾ cup at a time, stirring constantly and allowing each addition of broth to be absorbed before adding the next. With the final addition of broth, stir two-thirds of the sautéed mushrooms into the risotto. Cook until the risotto is creamy and the rice has a slight al dente bite. Stir in the cheese and season to taste.

Top with the remaining sautéed mushrooms, garnish with parsley, and serve with more grated cheese, if desired.