Place the stock and polenta in a medium-sized saucepan and bring to a boil, stirring often.
Reduce heat to low and stir in the milk and salt. Stir until starting to thicken, about 15 minutes. Remove from heat and stir in the butter and Parmesan cheese.
If making ahead of time, reheat over medium heat and add more stock, stirring constantly, until creamy (up to 1 to 2 cups may be necessary).
If making ahead of time, reheat over medium heat and add more stock, stirring constantly, until creamy (up to 1 to 2 cups may be necessary).
Heat the oil in a large skillet over medium-high heat. Once shimmering, add the butter and stir with a wooden spoon until melted.
Stir in the onions and sauté, stirring frequently, until they have softened and started to turn a light brown.
Stir in the mushrooms and continue cooking for another 15 minutes. Stir in the garlic, thyme, and pinch of salt and pepper, and cook for 30 seconds more.
Add the balsamic vinegar and stir to coat. Stir in the red wine and stock and bring to a boil over high heat. Decrease the heat to medium and simmer until slightly reduced, about 5 more minutes.
Serve by adding a layer of the creamy polenta in small bowls and then spoon the mushroom mixture over the top. Garnish with a pinch more of thyme.
4 servings