To Make Pesto: With a food processor, pulse the nuts and garlic until minced. Add the oil and pulse until just blended. Add the basil, cheese, and salt, process until it is finely minced. Store in an airtight container.
To Make Chili: Heat the Melt in a Dutch oven over medium-high heat. Add onion and chicken and saute 5 minutes. Add carrot, pepper and celery, saute 4 minutes. Add chiles and next 6 ingredients and bring to a boil.
Cover, reduce heat, and simmer 25 minutes. Stir in 3 tablespoons of the Classic Pesto before serving.