Cut bread into bite-sized pieces and dry out via low heat oven or on countertop for a few days
Preheat oven to 350°F and add butter to a medium baking dish.
In a large bowl, toss bread with 3 tablespoons melted butter. Spread in an even layer on a baking sheet and toast until golden and crisp, 10 to 12 minutes.
In a large skillet over medium heat, add sausage. Cook, stirring occasionally, until no pink remains, 10 minutes. Season with salt and pepper and remove sausage with a slotted spoon. Melt remaining tablespoon of butter in pan.
Add onion and celery and cook until soft, 6 minutes. Stir in garlic, parsley, sage, thyme, and rosemary and cook until fragrant, 1 minute more. Season again with salt and pepper.
Place bread in a large bowl and add skillet mixture and sausage. Toss to combine.
In a medium bowl, whisk together chicken broth and beaten egg and pour over bread mixture. Season generously with salt and pepper and toss until completely coated.
Transfer mixture to prepared baking dish and cover with foil. Bake until cooked through, 35 minutes, then remove foil and cook until bread is crisp on top, 15 to 20 minutes more.