Stacked Polenta and Mushroom Casserole

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6 cups chicken stock, plus more if needed

2 1/2 cups coarse polenta 

Salt and coarse black pepper

Salt and coarse black pepper 

6 tablespoons butter, plus more for dotting casserole

1/2 cup dried porcini mushrooms (a couple of handfuls)

About 1/3 cup EVOO 

1 1/2 pounds cremini or white mushrooms, sliced 

2 tablespoons finely chopped fresh thyme

2 shallots, finely chopped 

1/2 cup dry white wine or 1/4 cup Marsala or sherry 

2 cloves garlic, grated 

1/4 cup flour 

2 1/2 cups milk 

Freshly grated nutmeg, as needed 

1 1/2 cups grated Parmigiano-Reggiano