Preheat oven to 350°.
Cook chicken sausage in a skillet then set aside in a large bowl.
Add butter to that same skillet, then cook the leeks and celery over medium high heat for a few minutes until soft, then add the thyme and stir.
Pour in the chicken broth and heavy cream, then add salt and pepper. Simmer for 5 minutes, then pour mixture into the bowl with the chicken sausage. Now, add the eggs to that bowl and stir. Pour mixture into a 9x13 baking dish, then add in the cornbread and GENTLY stir! Some if it is going to break and that's ok--you want there to still be a handful of full cornbread cubes.
Bake stuffing for 45 minutes or until cornbread is golden on top. If cornbread begins to burn, add foil over top of the casserole dish.
Remove, season with additional salt and pepper and enjoy!