In a cast-iron pan, sauté the onion, celery, and carrots till translucent, about five minutes. Let cool while adding other ingredients to your crockpot. Add the chicken, roasted garlic, seasonings, green chilies, salsa and 4 cups of the chicken broth to the crock pot and turn on low heat. Add sautéed vegetables and stir mixture to combine. Add the white beans. Cover while making the cheese sauce.
Return the cast-iron pot to the cooktop. On low heat, melt butter, then add flour, stirring constantly with a whisk. Cook for about two minutes until bubbling and thickened. Slowly add the remaining cup of chicken stock. Whisk until completely combined and thickened. Add the can of evaporated milk. Stir until well combined. Slowly add the grated cheese and whisk until completely incorporated and the consistency is smooth. Pour cheese sauce into the crock pot. Gently stir to combine and cook over low heat until ready to serve.
Garnish bowls with chopped cilantro, salsa, avocado, and grated cheese. Serve with tortilla chips, cornbread or cheese bread.