Preheat the oven to 350°. In a very large bowl, toss the the bread cubes with the 4 tablespoons of melted butter, then spread on 3 large baking sheets. Sprinkle with the Parmesan and bake for 15 to 20 minutes, or until browned and crisp. Let cool. Return the bread to the bowl. Leave the oven on.
Melt the remaining 1 stick of butter in a large skillet. Add the onion, scallions and celery and cook over moderately low heat, stirring, until softened, about 12 minutes. Add the ham and cook over moderately high heat, stirring, until golden, about 4 minutes. Stir in the apples and thyme and let cool.
Scrape the apple mixture over the bread cubes and toss to coat; season with salt and pepper. In a large bowl, whisk the stock with the eggs and pour over the stuffing; mix thoroughly.
Butter 2 large, shallow baking dishes and divide the stuffing between them. Cover with foil and bake until heated through, about 25 minutes. Uncover and bake until browned and crisp, about 15 minutes. Serve at once.
Make Ahead: The stuffing can be assembled and refrigerated overnight. Bring to room temperature before baking.
Notes: Ground cooked country ham is available from Meacham Country Hams, 800-552- The prosciutto can be finely ground in a food processor.