Preheat the oven to 375°F. Put the mushrooms in a 2 cup liquid measuring cup and fill it with hot water. Dice the bread into rough 1-inch cubes. Smear the interior of a large shallow baking dish (at least 9 × 13 inches) with 2 tablespoons of the butter.

Lift the dried mushrooms out of their soaking liquid so that any grit they have released stays at the bottom of the cup. Chop them finely. Pour the mushroom soaking liquid through a fine strainer, agitating it as little as possible and keeping the last ½ cup or so of liquid behind.

Melt the remaining butter in a large skillet over medium heat and cook the onion in it until it softens, about 5 minutes. Add the sliced button mushrooms and the chopped porcini and cook them for another 5 minutes. Pour in the strained liquid and simmer for another 5 minutes. Stir in the marjoram and turn off the heat.

Whisk the eggs, milk, salt, and pepper together in a very large mixing bowl. Stir in the cooked mushrooms and onion. Add the bread cubes and gently toss them with a rubber spatula. Pour the pudding into the buttered baking dish. Bake for 50 to 55 minutes, or until browned on the top and firm to the touch in the center.