Preheat the oven to 350 degrees F (175 degrees C). Grease an 11x15-inch baking dish with nonstick spray.
Place chicken breasts in a large saucepan with enough water to barely cover; bring to a boil over medium heat. Reduce heat to medium-low and simmer until chicken is no longer pink in the center, about 15 minutes. Remove from the saucepan, drain, and shred with 2 forks.
Place shredded chicken into a large bowl. Add 1 (8-ounce) package Colby-Monterey Jack cheese and canned green chilies. Mix well.
Spoon about 3 tablespoons chicken mixture in a line down the center of each tortilla and fold tortilla in half, enclosing filling. Starting at the filled side, roll the folded tortilla into a cylinder. Place each enchilada, seam-side down, into the prepared baking dish.
Whisk together condensed soup, half-and-half, chicken broth, and about ⅓ of the remaining Colby-Monterey Jack cheese package in another large bowl until well combined. Pour over enchiladas, then sprinkle remaining Colby-Monterey Jack cheese over the top. Let stand for 10 to 15 minutes.
Bake in the preheated oven until bubbly and cheese topping is melted and beginning to brown, about 25 minutes.
