Preheat the oven to 350 degrees F. Melt 2 tablespoons butter in a large nonstick skillet over medium-high heat. Add the mushrooms and cook, stirring occasionally, until tender and lightly browned, about 6 minutes. Push the mushrooms to one side of the skillet and add the remaining 2 tablespoons butter, then add the scallion whites, shallot, thyme, 1 teaspoon salt and a few grinds of pepper. Cook, stirring, until the shallot is softened, about 3 minutes. Remove the skillet from the heat and carefully add the brandy. Return to the heat and cook until the pan is dry, about 30 seconds. Let cool slightly.
Whisk the eggs, heavy cream, chicken broth, garlic, 1 teaspoon salt and a few grinds of pepper in a large bowl. Stir in the bread cubes, mushroom mixture, 1 cup each fontina and cheddar, the scallion greens and parsley. Let sit 30 minutes to allow the bread to soak up the egg mixture.
Butter a 3-quart baking dish. Add the stuffing and top with the remaining 1/4 cup each fontina and cheddar. Bake until golden brown, about 1 hour. Let sit about 20 minutes before serving.
Stale bread is best for stuffing. If yours is fresh, cut it into cubes, spread on a baking sheet and bake in a 325 degrees F oven until dried out, about 20 minutes.