Chicken Chili

1/3 cup vegetable oil
1 large onion, chopped
2 garlic cloves, minced
1 large jalapeño, seeded and minced
1 cup fresh or frozen corn kernels
2 tablespoons New Mexico chile powder
1 tablespoon ground cumin
Salt
3 tablespoons tomato paste
5 cups shredded chicken (from 2 rotisserie chickens)
3 cups low-sodium chicken broth
2 tablespoons chopped cilantro

Cornbread

1 1/2 cups coarse cornmeal
1 cup plus 2 tablespoons all-purpose flour
3 tablespoons sugar
2 teaspoons baking powder
1 teaspoon salt
6 ounces cheddar cheese, shredded (2 cups)
1 cup plus 2 tablespoons milk
1/4 cup plus 2 tablespoons vegetable oil
3 large eggs, beaten