In a large cast iron skillet over medium-high heat, melt the butter, then add the onion and celery. Cook until soft and translucent, about 6–8 minutes.
In a large bowl, combine the chicken broth, eggs, sage, parsley, salt and pepper. Fold in bread crumbs until the liquid is absorbed. Add the bread mixture to the skillet and combine with the vegetables.
Grill over indirect medium heat, 350º to 450º F, with the lid down until the top is browned and the dressing cooked through, approximately 30–40 minutes.