Melt 4 tbsp butter in a large skillet over medium heat. Add the celery and onion and cook for 8 minutes until the onion is translucent and the celery has softened. Remove from heat and cool for 5 minutes.

Crumble the cornbread and tear the sandwich bread into 1-inch pieces into a large mixing bowl, leaving a mix of fine and coarser pieces.

Add the cooled vegetables, chicken broth, cream of chicken soup, sour cream, beaten eggs, sage, thyme, black pepper, and salt to the bowl. Stir until no dry pockets remain. The mixture should look wet and loose.

Preheat the oven to 375°F (190°C). Grease a 9x13-inch baking dish, pour in the dressing, and spread evenly. Bake uncovered for 45 to 55 minutes until the top is deep golden brown and the center reads 165°F (74°C).

Let the dressing rest for 5 minutes, cut into squares, and plate. Spoon pan drippings or warm broth over each square and finish with a pinch of fresh thyme leaves.