Preheat the oven to 350 degrees F. Coat a 9-by-13-inch baking dish with cooking spray. Spread the challah and cornbread on separate baking sheets and bake until lightly toasted and dry; transfer to a large bowl.
Heat the butter in a large skillet over medium heat. Add the onions, garlic and celery and cook until the onions are translucent, 10 to 12 minutes. Add the sage, parsley and 2 cups chicken broth and bring to a boil. Add salt to taste and 1 teaspoon pepper.
Pour the broth mixture over the bread in the bowl and mix with a wooden spoon until combined. Whisk the remaining 1/4 cup chicken broth with the egg in a small bowl; add to the stuffing mixture and toss to combine. Spoon the stuffing into the prepared baking dish, pressing it down gently so it is evenly distributed. Bake until dry on top and golden brown, about 1 hour.