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Cover the dried porcinis with the remaining 1 cup chicken stock and 1/2 cup water in a small pot; bring to a simmer and cook to reconstitute. Remove the tender mushrooms and chop. Reserve the mushrooms and their liquid.
Heat the EVOO in a large, deep skillet. Add the fresh mushrooms and cook until dark and tender, about 15 minutes. Add the thyme and shallots, season with salt and pepper and cook, stirring, for 2 to 3 minutes. Deglaze the skillet with the wine. Add the stock and porcinis with some of their liquid (be careful to avoid any grit) and reduce the heat to low.
To make the bechamel, melt the remaining 4 tablespoons butter in a saucepot, add the garlic and swirl a minute or two. Whisk in the flour and cook for 1 minute. Whisk in the milk, season with salt, pepper and nutmeg and cook until thickened. Add half of the cheese.
Place a layer of the polenta in a medium casserole dish and top with half of the mushrooms and bechamel. Repeat the layers, and then end with a layer of polenta. Dot with butter and top with the remaining cheese.
Cool and store for a make-ahead meal, or bake to serve. Bake at 375 degrees F from room temperature until brown, bubbling and hot through, about 50 minutes.