Cover the chicken breast halves with water in a saucepan. Boil for 20 minutes. Drain and reserve.
Preheat the oven to 375°F. Debone the chicken, discard skin and bones. Chop the chicken meat and reserve.
Stir green chilies, jalapeño, and garlic into a skillet of heated oil at medium temperature. Cook until fragrant then add cream cheese and half the Monterey Jack. Gradually add water as cheese melts. Stir chopped meat into skillet and remove from heat.
Spoon the chicken mixture into tortillas, and roll up.
Place enchiladas seam side down in 9x13 baking dish. Sprinkle remaining Monterey Jack over top; pour cream over all.
Bake for 30 minutes until golden brown.