Preheat the oven to 350 degrees F (175 degrees C) and grease a baking dish.

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Remove casings from the chorizo and cook in a large skillet until crumbly and browned, 7 to 9 minutes. Remove from the skillet, add to a large bowl, and set aside.

Melt butter in the same skillet. Add onion and celery and cook until soft, 5 to 7 minutes. Add garlic and cook until fragrant, about 1 minute. Add cilantro and transfer mixture to the bowl of chorizo. Add cornbread and chicken broth; stir until well incorporated. Transfer to the prepared baking dish.

Bake in the preheated oven until thoroughly warmed and stuffing starts to brown, 20 to 25 minutes. Serve garnished with cotija cheese.