Place the boneless, skinless chicken breasts or thighs into the slow cooker. Pour the green enchilada sauce and chicken broth over the chicken, ensuring the poultry is fully submerged in the liquid.

Cover the slow cooker and cook on low heat for 6 to 8 hours, or until the chicken is tender and can be easily shredded.

Once cooked, remove the chicken from the cooker and shred it using two forks. Return the shredded meat to the broth.

Incorporate the Monterey Jack cheese, cubed cream cheese, half and half or Greek yogurt, and green salsa into the slow cooker. Stir well to blend thoroughly.

Continue to cook on the low setting for an additional 30 minutes, stirring occasionally until the soup reaches a creamy and smooth consistency.

Taste the mixture and adjust salt and black pepper as needed before serving.

Ladle the creamy chicken mixture into bowls and top with your preferred garnishes, such as sliced avocado, fresh cilantro, sliced green onion, and a dollop of sour cream.