Add the dried porcini mushrooms to a cup of hot water and soak until softened, about 15 minutes. Using a slotted spoon, transfer the mushrooms to a cutting board and finely chop; set aside. Carefully pour remaining mushroom liquid into a large bowl, leaving any grit behind.

In a large (12-inch) skillet or Dutch oven, heat 1 tablespoon oil and 1 tablespoon butter over high. Add fresh mushrooms and cook, stirring occasionally, until browned. Lower heat to medium. Add sage, shallots and chopped porcini, and cook, stirring, until shallot is translucent, 3 to 4 minutes. Add cognac, if using, season to taste with salt and pepper and cook, stirring, 1 minute more. Transfer to a plate.

In the same pan, melt 1 tablespoon butter over medium heat. Add the leeks and a big pinch of salt and cook, stirring, until wilted, 5 to 7 minutes. Return the mushroom mixture to the pan and stir to combine with the leeks. Remove from heat.

Add eggs to the large bowl with the mushroom liquid, and whisk to blend. Add cream, milk, cayenne, 1½ teaspoons salt and plenty of black pepper. Whisk thoroughly to combine, then add the Gruyère.

Grease the bottom and sides of a 9-by-13-inch baking dish with butter, then add about ¼ of the fresh mushroom mixture and arrange in an even layer. Arrange the bread slices on top in an overlapping pattern. Sprinkle the remaining mushroom mixture over top, tucking it in between the bread slices.

Ladle the custard mixture over top, evenly distributing the cheese. Refrigerate, covered, overnight (see Tip below).

When ready to cook, remove the dish from the refrigerator about 30 minutes before baking. Heat the oven to 350 degrees. Top the pudding with Parmesan and bake for 40 to 45 minutes until set in the center (it may puff a bit). Run it under the broiler for 2 to 3 minutes to brown the top, watching carefully. Let sit 10 minutes, then serve warm, topped with additional sage if desired.

Tip: If you don't have time to soak overnight, let sit for at least 15 minutes, pressing down on the bread from time to time to encourage absorption. Before baking, tear into a piece of bread to make sure the custard has fully permeated it — the bread shouldn’t look dry at the center — and let sit another 15 minutes, if needed, to allow the custard to fully soak in. Decrease the cooking time by 10 to 15 minutes.