Preparation

Make the filling

• Heat the oil or melt the butter in a large Dutch oven over medium-high heat. Add the chorizo and cook, stirring occasionally to break up the pieces, until browned, about 7 minutes. Remove the chorizo using a slotted spoon and reserve, leaving the fat behind. Add the celery, onion, bell pepper, and jalapeño, and cook, stirring, until soft and translucent, 7 to 8 minutes. Stir in the garlic.
• Add the flour, chili powder, cumin, oregano, and coriander, and cook for 1 minute, stirring constantly. Stir in the chicken broth about a third at a time, allowing it to thicken a little before adding the next third. Bring the liquid to a simmer, then add the reserved chorizo, chicken, beans, corn, carrot, and 1/4  tsp. salt.
• Return the mixture to a simmer and cook, covered with the lid ajar, for about 30 minutes, stirring often. If the chili seems thin, simmer uncovered for about 15 minutes to thicken. Season to taste with salt and pepper, then remove from the heat.
• Lightly oil or butter a 3- to 3-1/2-quart casserole dish. Place on a large rimmed baking sheet. Carefully transfer the chili to the casserole and cool for 30 minutes.
• Position a rack in the center of the oven and heat to 400°F.

Make the topping

• Whisk the flour, cornmeal, sugar, baking powder, and salt in a medium bowl. Make a well in the center and add the milk, egg, and oil. Whisk the liquids to combine, then whisk the dry and wet ingredients. Stir in the corn.
• Pour the batter evenly over the chili. Transfer to the oven and bake until the cornbread is golden and the chili is bubbling, about 30 minutes. Transfer to a wire rack and cool for at least 20 minutes before serving.