Preheat the oven to 350°F. Lightly grease a 9x13-inch baking dish with butter.
Cut the bread into 1/2-inch cubes (around 12 cups total) and spread evenly across 2 large baking sheets. Transfer to the oven and bake until the bread is completely dry and beginning to crisp, about 20 minutes. Remove from the oven and set aside.
Meanwhile, heat a large skillet over medium heat and melt 9 tablespoons of butter. Add the onions, celery, garlic, thyme, 1 teaspoon salt, and 1/2 teaspoon pepper. Cook, stirring often, until the onions and celery are very tender, 8 to 10 minutes. Transfer to a large mixing bowl and set aside.
Increase the heat to medium-high. Add the sausage and sage. Cook, breaking up the meat with a wooden spoon, until the sausage is cooked through, about 6 minutes.
Add the remaining 1 tablespoon of butter and toss until coated. Add the wine, stir, and cook until reduced by half, 2 to 3 minutes. Transfer the sausage mixture to the bowl with the onions and celery.
Add the toasted bread to the mixing bowl and toss until well combined. Add the chicken broth, eggs, parsley, dried sage, dried thyme, the remaining 1 teaspoon salt, and 1/2 teaspoon pepper. Toss again to combine.
Transfer into the prepared baking dish and bake, uncovered, for 20 to 25 minutes until the bread is golden brown.