In a 10- to 12-inch frying pan over medium heat, stir chorizo until crumbly and browned (leave some large chunks), about 5 minutes. Transfer to paper towels to drain.
Return pan to medium-high heat and add butter. When melted, add onion, celery, carrot, and garlic. Stir until vegetables are lightly browned, about 10 minutes. Add cilantro, thyme, oregano, and sage; stir until fragrant, 1 minute. Add chorizo and 2 cups chicken broth and cook just until boiling. Remove from heat.
In a large bowl, combine cornbread cubes and vegetable-chorizo mixture. Stir until evenly moistened, seasoning to taste with salt and pepper. If mixture is too dry for your liking, moisten with additional 1/2 cup chicken broth. Spoon into a 9- by 13-inch baking dish.
Cover and bake in a 450° oven for 10 minutes; uncover and continue to bake until top is browned and dressing is heated through, 10 to 15 minutes longer. Serve hot.