Melt butter in a large saucepan over medium heat; cook and stir carrot, celery, onion and garlic into hot butter until slightly softened, about 2 minutes.

Stir flour into mixture to make a paste; cook until lightly browned and flour gives off a slightly toasted smell, about 5 minutes. Watch carefully, flour burns easily.

Remove saucepan from heat and set aside to cool, about 15 minutes.

Stir corn, beans, chicken, and 3 cups of chicken broth in a large soup pot over medium heat. Whisk remaining 1 cup chicken stock into vegetables and flour mixture until thoroughly combined. Stir mixture into ingredients in soup pot. Bring to a simmer, stirring constantly, until thick, about 5 minutes.

Stir half-and-half, cilantro, cumin, salt, and black pepper into chowder. Bring back to a boil, reduce heat to low, and simmer for 15-20 minutes.