Heat the oil in a large skillet over medium heat. Add the sausage and cook, using a wooden spoon to break it up, until browned and cooked through, about 10 minutes. Using a slotted spoon, transfer the sausage to a medium bowl, leaving the sausage fat in the skillet. Add the mushrooms, celery and onions to the skillet and saute, stirring occasionally, until tender, about 10 minutes. Stir in the sage and cook 2 more minutes. Season with salt and pepper. Turn off the heat. Place the cubed cornbread in a large bowl and add the sauteed vegetables and sausage and stir to combine. Set to the side and let cool slightly. Whisk the chicken broth, milk, egg, 1/2 teaspoon salt and 1/2 teaspoon pepper in a medium bowl. Pour over the cornbread mixture and gently stir to coat. (If not baking right away, refrigerate until ready to bake.)
Preheat the oven to 375 degrees F.
Transfer the stuffing mixture to a greased 9-by-13-inch baking dish. Bake until golden brown, 30 to 40 minutes.