Cook chicken in water and salt for 45 minutes to 1 hour. Remove and cool. Strain and reserve 1/2 cup broth. When chicken is cool, cut into chunks.

Melt butter in saucepan. Add onion, garlic, and cumin; cook until onion is tender. Stir in flour. Gradually add broth. Cook and stir until thickened and smooth. Add green chilies. Stir in half and half. Season with salt. Add reserved chicken and bring mixture to a simmer. Drop tortillas into simmering chicken mixture. Cover and turn off heat. Let stand, covered, for 15 minutes.