Generously grease a 9×13 baking dish with nonstick cooking spray or butter. Set aside.
Heat a large skillet over medium heat, add breakfast sausage, and break it into chunks with a wooden spoon. Let it cook while stirring for 3-5 minutes.
Add about a tablespoon of butter to the pan with the sausage, followed by onions, celery, garlic, Italian herbs, sage, and thyme. Then cook until the vegetables are softened, 3-5 minutes. Remove from heat.
In a large bowl, combine day-old cornbread and bread cubes.
Fold in the sausage and vegetable mixture, including the flavorful pan juices.
Mix the cans of cream of mushroom soup and cream of chicken soup, then mix them gently into the bread mixture. Add the lightly beaten eggs and mix them in.
Then add the reserved chicken broth gradually until the dressing is moist and sponge-like, not soupy. Taste and adjust seasoning.
Spread roughly half of the cornbread dressing in the prepared baking dish.
Distribute the shredded chicken over the dressing in an even center layer.
Cover the chicken with the remaining cornbread dressing. Dot the top with the remaining butter.
Bake at 350°F (180°C) for 35-40 minutes, until hot throughout and lightly crusty and golden on top. Begin checking around 30 minutes.