Melt butter in large pot over medium heat. Add ham and sauté until browned, about 8 minutes. Using slotted spoon, transfer ham to extra-large bowl. Add onions and next 6 ingredients to drippings in pot. Cover and cook until vegetables begin to brown, stirring often, about 15 minutes; add to ham. (Can be made 1 day ahead. Cover and chill. Reheat before continuing.)
Stir all bread into ham mixture; add 31/4 cups broth. Season stuffing to taste with salt and pepper; mix in eggs.
To bake stuffing in turkey: Loosely fill main cavity and neck cavity of turkey with stuffing. Add enough broth to remaining stuffing to moisten slightly (1/4 cup to 3/4 cup, depending on amount of remaining stuffing). Generously butter baking dish. Spoon remaining stuffing into prepared dish. Cover dish with buttered foil, buttered side down. Bake stuffing in dish alongside turkey or while turkey is resting until heated through, about 25 minutes. Uncover stuffing in dish. Bake until top of stuffing is slightly crisp and golden, about 15 minutes longer.
To bake stuffing in dish: Preheat oven to 350°F. Generously butter 15x10x2-inch glass baking dish. Add enough extra broth to stuffing to moisten (3/4 cup to 1 1/4 cups). Transfer stuffing to prepared dish. Cover with buttered foil, buttered side down. Bake until heated through, about 40 minutes. Uncover and bake until top is slightly crisp and golden, about 20 minutes longer.