Arrange the bread cubes in a single layer on a rimmed baking sheet and let dry, uncovered, overnight.
Heat 2 teaspoons oil in a 12-inch cast iron skillet over medium heat. Cook, stirring frequently, and breaking up the sausage with a wooden spoon, until browned and cooked through, 10 to 12 minutes.
Use a slotted spoon to scoop out the sausage and transfer to a very large bowl. Leave the sausage drippings in the skillet.
Add 1 tablespoon butter to the fat in the skillet and set over medium heat.
Stir in the onion, celery, and garlic. Season with salt and pepper, and cook, stirring frequently, until softened, about 10 minutes. If the onion and garlic begin to brown, reduce the heat to medium low.
Scoop the vegetables into the bowl with the sausage. Stir in the bread cubes and let the mixture cool for 10 minutes.
Prepare a grill for indirect heat (400°F), ideally with the cool zone in the center and burners on either side of the cool zone. (If your grill will only accommodate a cool zone on one side and the hot on the other, rotate the skillet when you uncover it.)
In a medium bowl, whisk the eggs, broth, Parmesan, and parsley together. Season with salt and pepper.
Pour the broth mixture over the stuffing and stir well to thoroughly combine everything.
Wipe out the cooled skillet and coat thoroughly with butter.
Scoop the stuffing mixture into the skillet and press gently to help it stick together. Cover with foil.
Grill over indirect heat until hot throughout, about 20 minutes. Uncover and continue to cook until the edges are crisped and golden brown, about 15 minutes more.
Serve directly from the skillet.