Place a rack in the upper third of the oven; preheat to 300 degrees . Scatter the cornbread on a rimmed baking sheet. Bake 25 minutes. Let cool; transfer to a large bowl.
Working over a gas flame or under the broiler, char the poblanos. Transfer to a paper bag, close it and let the poblanos steam for 15 minutes. Peel and discard the skins, then chop. Set the oven temperature to 375 degrees .
In a large skillet, melt 2 tbsp. butter over medium-high. Add the sausage; cook until browned, 5 minutes. Transfer to the bowl of cornbread. In the same skillet, melt the remaining 2 tbsp. butter, then add the poblanos, onion, celery and garlic. Reduce the heat to medium; cook, stirring, until the onion is soft, 5 minutes. Add to the cornbread.
In a bowl, whisk together the broth and the eggs; season. Pour over the cornbread mixture; toss. Transfer to a greased 3-qt. baking dish. Bake until the top is browned, 35 minutes.