In a 12-inch cast-iron skillet, melt ¾ cup butter over medium heat. Add rosemary sprigs; cook until butter starts to brown, 3 to 4 minutes. Discard rosemary; pour butter into a medium bowl. Whisk in buttermilk; whisk in 3 eggs, one at a time. Place skillet in oven to preheat.
In a large bowl, whisk together cornmeal, flour, 2 teaspoons salt, baking powder, and baking soda. Stir buttermilk mixture into cornmeal mixture just until combined. Carefully spread batter into hot skillet.
Bake until a wooden pick inserted in center comes out clean, 20 to 25 minutes. Remove cornbread from pan, and let cool completely on a wire rack. Reduce oven to 375°. Wipe skillet clean.
In same skillet, melt remaining ¼ cup butter over medium-high heat. Add celery, onion, and garlic; cook, stirring occasionally, until softened, about 5 minutes. Remove from heat.
In a large bowl, break cornbread into 1-to 2-inch pieces. Stir vegetable mixture, minced rosemary, poultry seasoning, pepper, and remaining 2 teaspoons salt into cornbread.
In a medium bowl, whisk together 1 cup broth and remaining 2 eggs; gently stir into cornbread mixture just until bread mixture is moistened. Gradually stir in remaining chicken broth as needed until mixture is moistened but not wet. Spoon cornbread mixture into skillet.
Bake until top is golden brown and dressing is firm to the touch, 45 to 50 minutes. Let stand for 10 minutes before serving. Garnish with rosemary, if desired.
Notes
The cornbread can be made a day ahead and stored in an airtight container.
Recipe by Southern Cast Iron at https://southerncastiron.com/browned-butter-rosemary-cornbread-dressing/