Place chicken in a pot and cover with water. Boil until the juices run clear, about 25 minutes. Drain and let sit until cool enough to handle, about 10 minutes. Shred meat.

Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a medium baking dish.

Set aside 1/4 cup Monterey Jack for topping enchiladas. Fill each tortilla with equal amounts chicken and remaining cheese. Roll tortillas to form enchiladas. Arrange enchiladas in the prepared baking dish, then cover with enchilada sauce.

Bake in the preheated oven for 30 minutes. Sprinkle reserved cheese over top and continue to bake until melted, about 5 more minutes.