Heat oven to 350 degrees. Greas a 9x13" bakig dish with butter. Soak and chop the mushrooms and strain the soaking liquid. Set aside 1/2 cup of the liquid for this recipe. While mushrooms soak, toast the bread cubes on a baking sheet in the oven until dry and crisp but not necesssarily golden, 5-10 minutes. Set aside to cool. Turn off the oven.
Heat 2 tablespoons butter in a large, heavy skillet over medium high heat until it begins to brown. Add leeks and sprinkle generously with salt and pepper (about 1/2 teaspoon each). Cook, stirring, until leeks are translucent, soft, and browned in places, 8-10 minutes. Stir in mushrooms and the reserved mushroom liquid. Raise heat to medium high and cook until liquid almost completely evaporates, 1-2 minutes. Remove from heat.
Beat eggs in a large bowl. Whisk milk, cheese, 1/2 teaspoon salt, and a few grinds of pepper. Arrange half of bread cubes evenly over bottom of buttered baking dish. Scatter half of the mushroom-leek mixture. Pour over half of the egg mixture. Top with the remaining bread, and then the remaining mushroom-leek mixture. Sprinkle with salt and pepper. Cover and refrigerate over night.
Heat oven to 350 degrees. Let the strata sit at room temperature while the oven heats. Uncover and bake until eggs are set and top browns in places, about 45 minutes. Let rest for 10 minutes before serving.