In your slow cooker, Instant Pot, or large stockpot, place the boneless skinless chicken breasts or thighs. Pour in the green enchilada sauce and chicken broth, ensuring the chicken is submerged. If using a slow cooker, cover and cook on low for 6 to 8 hours until the chicken is tender. For the Instant Pot, set to high pressure for 8 minutes, followed by a quick release.

Once the chicken is cooked, carefully remove it from the cooker or pot using tongs. Let it cool for a minute before shredding with two forks, creating bite-sized pieces. Return the shredded chicken to the pot or slow cooker to allow the flavors to blend beautifully.

Incorporate the half and half or heavy cream, softened cream cheese, and Monterey Jack cheese. Stir well until the cheeses melt and the mixture becomes creamy and velvety. Adjust your pot's temperature to medium heat or set your Instant Pot to sauté mode for a few minutes to aid melting.

Once the cheese has melted, fold in the green salsa (salsa verde) to add brightness and acidity. Taste and season with salt and pepper as desired.

Ladle the soup into bowls and top with avocado slices, chopped cilantro, and a dollop of sour cream. Serve with tortilla chips.