Preheat oven to 375 degrees F (190 degrees C).

Bring a saucepan of salted water to a boil. Boil chicken breasts until no longer pink in the center and the juices run clear, about 10 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove chicken breasts to a cutting board to cool. Reserve and cool chicken broth.

Shred the cooled chicken breasts and put into a large bowl; add cheese cubes, diced tomatoes with green chile peppers, cream of chicken soup, and cream of mushroom soup and stir.

Individually dip about half the tortillas into the cooled liquid from poaching the chicken; arrange into the bottom of a large baking dish.

Spread the chicken mixture over the tortillas.

Individually dip the remaining tortillas into the water and arrange atop the chicken mixture. Sprinkle chili powder over the tortillas.

Bake in preheated oven until the top layer of tortillas begin to turn golden brown and crunchy, 30 to 45 minutes.