Spread 1/3 the potatoes into the bottom of the prepared baking dish in a layer; season with salt, black pepper, 1/2 the thyme leaves, and 1/2 the cooked mushrooms. Sprinkle almost half the sottocenere cheese over potatoes, reserving about 3 tablespoons for the top. Layer next 1/3 the potatoes over cheese, season with salt and black pepper, remaining 1/2 thyme leaves, 1/2 the remaining cheese, remaining mushrooms, and last layer of potatoes. Season potatoes with more salt and black pepper. Pour chicken stock and cream over the top. Sprinkle with more salt and black pepper and reserved 3 tablespoons of sottocenere cheese. Cover dish loosely with aluminum foil; tent the foil a little so it doesn’t touch the potatoes.