Heat oil in a Dutch oven over medium-high heat. Add onion and chicken; sauté 5 minutes. Add carrot, bell pepper, and celery; sauté 4 minutes. Add chiles and the next 6 ingredients (chiles though broth); bring to a boil.
Cover, reduce heat, and simmer 25 minutes. Stir in Classic Pesto.
Note: The chili and pesto can be made ahead and frozen for up to 3 months. Prepare and freeze 3 tablespoons Classic Pesto. Prepare the chili without Classic Pesto, and spoon into a freezer-safe container. Cool completely in refrigerator; cover and freeze. Thaw chili and pesto in refrigerator. Place chili in a large skillet; cook over medium-low heat until thoroughly heated, stirring occasionally. Stir in Classic Pesto.