Rinse chicken and pat dry with paper towels. Place chicken in a 3-quart saucepan, add water (or broth) and salt. Cover and simmer 45 minutes to 1 hour or until chicken is tender. Remove chicken from broth and set aside to cool. Strain broth and reserve.
When chicken is cool enough to handle, remove and discard skin and bones. Cut chicken into chunks and set aside.
Melt butter in a 3 quart saucepan. Add onion, garlic and cumin; cook, stirring until onion is tender. Stir in flour. Gradually add 3 cups of reserved broth (if necessary, add water to complete measure). Cook and stir until thickened and smooth. Add green chilies. Stir in half and half. Season with salt and pepper to taste. Add reserved chicken. Bring mixture to a simmer.
Cut tortillas into 2 x 1 inch strips and drop into the simmering mixture. Cover and turn off the heat. Let stand, covered 15 minutes. Garnish each serving with cheddar cheese, if desired.