Heat the oven to 350°F. Grease a 9 x13-inch baking dish with butter. Put the morels in a medium heatproof bowl. Pour in 2 cups boiling water and weight down the morels with a small plate so they are submerged. Soak until they're plumped and softened, about 20 min.
While the mushrooms soak, toast the bread cubes on a baking sheet in the oven until dry and crisp but not necessarily golden, 5 to 10 min. Set aside to cool and turn off the oven. Use a slotted spoon to transfer the mushrooms to a cutting board, squeezing any excess liquid from the mushrooms back into the soaking liquid. Let cool. Remove and discard any tough stems. Coarsely chop the morels. Strain the soaking liquid through a coffee filter or paper towel set in a sieve and reserve 1/2 cup (save the remainder for another use).
Heat the 2 Tbs. butter in a large, heavy skillet over medium heat until it begins to brown. Add the leeks and sprinkle generously with salt and pepper (about 1/2 tsp. each). Cook, stirring, until the leeks are translucent, soft, and browned in places, 8 to 10 min. Stir in the mushrooms and the reserved 1/2 cup mushroom liquid, raise the heat to medium high, and cook until the liquid almost completely evaporates, 1 to 2 min. Remove from the heat.
Beat the eggs in a large bowl. Whisk in the milk, Parmigiano, 1/2 tsp. salt, and a few generous grinds of pepper. Arrange half of the bread cubes evenly over the bottom of the buttered baking dish. Scatter half of the mushroom-leek mixture over the bread. Pour over half of the egg mixture. Top with the remaining bread, and then the remaining mushroom-leek mixture. Pour the remaining egg mixture over the strata. Sprinkle with salt and pepper. Cover and refrigerate for at least 4 hours or preferably overnight.
Heat the oven to 350°F. Let the strata sit at room temperature while the oven heats. Uncover and bake until the egg sets and the top browns in places, about 45 min. Let rest for 10 min. before serving.