Spread the bread cubes out on a baking sheet and toast in the oven until golden and crisp, about 5 minutes. Set aside to cool.
Trim the dry part from the bottom of the cremini stem and quarter. Stem and quarter the shiitakes. Melt 2 tablespoons of the butter with the olive oil in a large skillet over medium heat. Add the mushrooms, shallots, garlic, and rosemary and cook, stirring occasionally, until the vegetables soften and brown, about 25 minutes. Add an additional tablespoon of butter, if needed, to prevent sticking. Carefully add the Cognac (if working over gas, pull the skillet from the heat), and simmer until the liquid is slightly syrupy. Stir in the chicken broth and salt and bring to a simmer.
Whisk the cream, eggs, and pepper, to taste, in a large bowl. Add the bread, mushroom mixture, and parsley and toss until the bread is moistened. Transfer the dressing to a buttered 1 1/2-quart baking dish. Bake, uncovered, until the dressing sets and the top browns, about 1 hour. Let dressing sit for 15 minutes before serving.
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