Place the dried mushrooms in pot with 2 cups of the stock. Simmer to soften the mushrooms and flavor the stock. When rehydrated, remove the mushrooms from the stock and chop. Reserve the stock, except for the very bottom, which may contain grit from the mushrooms.
Sprinkle the chicken with salt and pepper, and then dredge in flour. Heat the oil in a pan over medium-high heat and add the chicken to brown on both sides. Remove to a plate and add a bit more oil if necessary. Add 2 tablespoons of the butter to the pan and melt. When the butter foams, add the garlic, onions, sage, thyme and some salt and pepper, and cook to soften, 7 to 8 minutes. Stir in 2 tablespoons flour, then deglaze the pan with the wine. Add the chopped mushrooms and whisk in the flavored stock. Cook to thicken a bit and add the chicken back to the sauce.
Cool and store for a make-ahead meal. Reheat over medium heat.
To serve, heat the remaining 1 cup stock, the milk and 1 cup water to a low boil. Whisk in the polenta, and then whisk to thicken, 2 to 3 minutes. Stir in the remaining 2 tablespoons butter, some salt and pepper and the grated cheese.
Serve the polenta in shallow bowls topped with the chicken, gravy and a sprinkle of parsley.