Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.

Bring a saucepan of lightly salted water to a boil. Cook chicken breasts in boiling water until no longer pink in the center and the juices run clear, 7 to 10 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove chicken to a cutting board to cool.

Beat cream cheese, garlic, cumin, salt, and pepper together in a bowl. Stir tomatoes, green onion, green chiles, cilantro, and lime juice into the cream cheese mixture. Add 1 cup Cheddar cheese and fold to integrate.

Shred cooled chicken with a pair of forks into strands; stir into cream cheese mixture. Spoon about ½ cup of the chicken mixture into the center of each tortilla and fold tortilla around the filling; arrange into the prepared baking dish. Top tortillas with remaining Cheddar cheese and green enchilada sauce.

Bake in the preheated oven until hot and the cheese is melted, about 20 minutes.