How to Make It
In a small bowl, soak the dried porcini in the hot water until softened, about 30 minutes. Drain and finely chop the mushrooms, discarding any tough bits.
Melt 2 tablespoons of the butter in a large skillet. Add the leeks and a pinch each of salt and pepper and cook over moderately high heat, stirring occasionally, until softened, about 5 minutes. Add the garlic and cook for 2 minutes longer. Add the remaining 2 tablespoons of butter along with the assorted mushrooms and chopped porcini and cook over high heat, stirring frequently, until the mushrooms have released their liquid. Pour in the wine and cook, stirring frequently, until the liquid has evaporated and the mushrooms begin to brown, about 10 minutes. Season the mushrooms with salt and pepper.
Preheat the oven to 375°. Lightly butter a 3-quart baking dish. Spread the diced bread in the baking dish and spoon the mushrooms on top. In a large bowl, whisk the eggs with the cream, 1 cup of the Parmesan cheese, the thyme and a pinch each of salt and pepper. Pour the custard over the mushrooms. Press a sheet of plastic wrap directly on the custard and let the bread pudding soak for 30 minutes.
Discard the plastic wrap. Sprinkle the the remaining 1/2 cup of cheese and the parsley on top of the pudding. Bake for about 40 minutes, or until the custard is set and the top is brown. Let stand for 10 minutes before serving.