Heat your oven to 375 degrees Fahrenheit. Place the crumbled cornbread in a large mixing bowl and set it aside. The bread should be broken into rough, uneven pieces, not fine crumbs. A mix of sizes gives you better texture in the finished dressing.

Melt 3 tablespoons of the butter in your 12-inch cast iron skillet over medium heat. Add the diced onion and celery and cook, stirring occasionally, for about 8 minutes until the onion turns soft and translucent and the celery is tender. Add the garlic and cook 1 more minute until fragrant. Stir in the sage, thyme, salt, and pepper.

Scrape the cooked vegetable mixture into the bowl with the crumbled cornbread. Add the beaten eggs and toss everything together gently. Pour in 2 cups of the warm broth and stir to combine. The mixture should look moist and hold together when you press a handful, but it should not be sitting in liquid. If it looks dry, add the remaining broth a splash at a time.

Melt the remaining tablespoon of butter in the same skillet over medium heat, swirling to coat the bottom and sides. Spoon the cornbread mixture into the skillet and press it down firmly with the back of a spoon so it makes contact with the bottom and sides of the pan. Those contact points are what create the crispy edges.

Bake uncovered for 35 to 40 minutes. The top should be golden brown and the edges should look set and pulling slightly from the sides of the skillet. The center should not jiggle when you tap the pan. Let it sit for 5 minutes before scooping and serving directly from the skillet.