Boil chicken, until meat falls from bone.
Meanwhile, as chicken boils, combine all soups, green chilis, and chopped onion in a large bowl.
Debone and remove skin if you prefer, tear chicken into bite size pieces. Add to soup mixture.
Layer 1/2 of the mixture in 13 x 9" baking or casserole pan.
Cut the tortillas into fourths and top the mixture with Monterey Jack cheese, then another layer of tortillas, another layer of cheese, then layer with remaining chicken and soup mixture, and layered tortillas, then top with remaining Monterey Jack cheese.
Bake in 375 F oven for 45 minutes or until lightly browned.
Let stand for 10-15 minutes. Cut into three or four serving sized portions and serve.
