Preheat oven to 350 degrees. Heat oil in a 12-inch skillet over medium heat. Add onion and celery; cook, stirring occasionally, until onion is translucent, 6 to 8 minutes. Add liver; cook, stirring, until beginning to brown, about 4 minutes.Advertisement
In a large bowl, combine onion mixture, cornbread, and sage.
Pour broth over mixture; stir just until combined. Season with salt and pepper. Transfer to a buttered 1 1/2-quart shallow baking dish.
Dot with butter. Bake until golden brown on top and crisp around edges, 40 to 45 minutes.
Note: If making ahead, prepare stuffing through step 3; refrigerate, covered.