Add oil and onion to a soup pot. Sauté on medium high heat, stirring frequently, until onion is tender. Then add minced garlic and sauté until golden in color, about one minute.
Add chicken stock and chicken to pot. Reduce heat to medium. Simmer until chicken is cooked through, about 15 minutes.
Carefully remove the chicken breasts from the pot. Shred or chop. Add the chicken back to the chili.
Add corn, beans, cream cheese, salsa verde, and seasonings to the soup pot. Stir to break up the cream cheese so it starts to mix with the other ingredients. Simmer until chili is warmed through, about 5 to 10 minutes.
Taste and add extra salt if desired. Ladle into bowls. Serve with crackers or tortilla chips.