Preheat the oven to 375 degrees F.
Heat the olive oil and 1/2 stick of butter in a large frying pan over medium heat. While the pan is heating, dredge the chicken pieces in the flour, tapping off any excess. Sear the chicken in the pan until golden on all sides, and then remove from the heat. Arrange the onions on the bottom of a baking dish, set the chicken on top and pour over the wine. Cover and bake for about 1 hour.
Transfer the chicken to a separate dish. Pour the wine and drippings from the baking dish reserves into a sauce pan, stir in the mushrooms and bring to a boil over medium heat. Reduce the heat to a simmer and cook until thickened, about 20 minutes. Whisk in the remaining butter and season with salt and pepper.
To serve, portion out the chicken and Polenta onto plates and top with the reduced red wine sauce and Parmigiano-Reggiano.
Bring 6 cups of water to a boil in a large sauce pan, and then add the salt. Gradually, and carefully, whisk in the polenta. Reduce the heat and cook until the mixture thickens and the cornmeal is tender. Turn off the heat, whisk in the butter and transfer to a serving dish.
The polenta can be cooked in 6 cups of vegetable or chicken stock instead of 6 cups of water. If using stock, omit the salt.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.