Cut the cornbread into 1-inch (2.5-cm) squares. Divide the cornbread between 2 rimmed baking sheets. If drying the cornbread overnight, leave the cornbread on the baking sheets, uncovered, at room temperature for at least 12 hours. If drying the cornbread in the oven, slide the baking sheets into the oven and bake, stirring occasionally, until dry, 50 to 60 minutes. Let the cornbread cool completely, about 30 minutes. In a 12-inch (30-cm) skillet over medium-high heat, cook the sausage, using a wooden spoon to break it up as it cooks, until no trace of pink remains, 7 to 15 minutes. Using a slotted spoon, transfer the sausage to a paper towel-lined plate.
Pour off all but 2 tablespoons fat from the skillet. Reduce the heat to medium and add the onion, celery, and 2 tablespoons butter to the fat in the skillet and cook until the vegetables soften, 5 to 6 minutes.
Stir in the garlic, sage, and thyme and cook until fragrant, about 30 seconds. Stir in the broth, remove from the heat, and let cool for 5 minutes.
In a large bowl, whisk together the half-and-half, eggs, salt, and cayenne, if using. Slowly whisk in the warm broth mixture until incorporated. Gently fold in the cornbread and sausage. Let the mixture sit, stirring occasionally, until the cornbread is completely saturated, about 20 minutes.
Preheat the oven to 375°F (190°C) and adjust the oven rack to the upper-middle position. Butter a 13-by-9‑inch (33-by-23-cm) baking dish.
Spoon the cornbread mixture into the prepared dish. Melt the remaining 4 tablespoons butter and drizzle it evenly over the cornbread.
Bake the dressing on the upper rack until the surface is golden brown and slightly crisp, 30 to 40 minutes. Let cool for 15 minutes before serving.