Special equipment: a 12-cup muffin tin
Preheat the oven to 300 degrees F.
Spread the bread and cornbread out on two rimmed baking sheets. Bake until dried out but not yet golden, 10 to 15 minutes. Transfer to a large bowl.
Lightly butter the cups of the muffin tin. Adjust the oven to 400 degrees F.
Melt the 2 tablespoons butter in a large skillet over medium-high heat. Add the sausage and cook until browned, breaking up any clumps, about 5 minutes. Add the celery, onion and cayenne, and cook, stirring occasionally, until soft, about 6 minutes. Transfer the mixture to the bowl with the bread and toss to coat. Let cool to room temperature. Add the egg and sage, and stir to combine. Add the chicken stock, 1 teaspoon salt and 1/4 teaspoon pepper; toss until well combined.
Divide the mixture evenly between the muffin cups, pressing to mound the tops. Bake until golden brown on top, about 35 minutes. Let cool in the pan for 5 minutes. Serve hot.
The muffins can be made up to 4 hours ahead and allowed to cool completely in the pan. Cover loosely with foil and reheat in oven before serving.