Preheat and prep dish: Preheat the oven to 375°F and grease a 9x13-inch baking dish with butter or oil so the enchiladas don't stick as they bake.
Cook the chicken: If starting with raw chicken, simmer breasts in chicken broth for about 15 minutes, until the centers lose their pink color and the juices run clear.
Shred the chicken: Remove the chicken from the broth and shred it with two forks while still warm, so the meat pulls apart into even strands. Reserve the broth for the sauce.
Cook the aromatics: Melt butter in a skillet over medium heat, then add diced onion and minced garlic. Cook for about 2 minutes, stirring often, until soft and fragrant.
Build the sauce: Pull the skillet off direct heat for a minute, then whisk in cream cheese, reserved broth, sour cream, green chiles, cumin, salt, and pepper until smooth and pourable.
Combine the filling: In a bowl, mix the shredded chicken with 1 cup of the Monterey Jack cheese and about half of the sauce, stirring until evenly coated.
Warm the tortillas: Warm the low carb tortillas for about 20 seconds in a damp paper towel in the microwave, or a few seconds per side in a dry skillet, until soft and pliable.
Fill and roll: Spoon about 1/3 cup of the chicken filling down the center of each warmed tortilla, roll snug, and place seam-side down in the prepared baking dish.
Top with sauce and cheese: Pour the remaining sauce evenly over the rolled tortillas, then scatter the remaining 1 cup of Monterey Jack cheese across the top.
Bake until bubbly: Bake for 20 to 25 minutes, until the edges turn golden and the sauce bubbles around the sides of the dish.
Garnish and serve: Let the enchiladas rest for 5 minutes, then sprinkle with chopped cilantro before scooping onto plates.