Warm the olive oil in the large skillet over high heat. Sauté the onions until soft, about 4 minutes, then add the mushrooms and sauté until soft, about 4 to 6 minutes. Stir in the thyme leave, oregano, garlic, and vegetable broth. Keep warm.
In the 4-quart pot, bring the milk and vegetable broth to a boil. Stir in the polenta and cook while stirring until the polenta absorbs the liquid, about 4 minutes. Stir in half of the Parmesan and the kosher salt.
Preheat the broiler. Divide the polenta among the gratin bowls and sprinkle the remaining Parmesan on top. Broil until the cheese melts and browns, about 5 to 6 minutes. Top each bowl with mushroom mixture and serve hot.