Preheat oven to 375 degrees. Spray a 9×13-inch baking dish with nonstick cooking spray. Set aside.
In a large skillet over medium-high heat, cook Italian sausage until browned and no longer pink, breaking up the meat into medium chunks.
Using a slotted spoon, remove sausage to a large bowl.
Add the cornbread and sourdough cubes.
Drain remaining fat from skillet and add the 1/2 cup butter. Melt butter over medium-high heat.
Add the celery, red onion, salt, and pepper. Cook for 5-7 minutes until slightly softened and translucent.
Add the garlic and continue to cook for 1 minute. Add vegetable mixture to sausage/bread mixture.
In a medium bowl, whisk together one cup chicken broth, heavy cream, eggs, parsley, and thyme leaves.
Pour mixture over bread mixture and toss lightly until well mixed.
Pour entire mixture into prepared dish. Drizzle with remaining 1/2 cup broth. Cover dish tightly with foil and bake for 30 minutes.
Uncover and bake for 30 minutes more, until lightly browned and bubbling. Let stand 10 minutes before serving.