Preheat oven to 375°F (190°C). Spray a 9x13-inch baking dish with nonstick cooking spray.
In a large bowl, combine the shredded chicken, green chiles, 1 1/2 cups of the Monterey Jack cheese, garlic powder, cumin, and a pinch of salt and pepper. Mix well and set aside.
In a saucepan over medium heat, melt butter. Whisk in flour and cook for 1 minute until lightly golden. Gradually whisk in chicken broth until smooth. Bring to a simmer and cook until slightly thickened, about 2-3 minutes. Reduce heat to low, then stir in sour cream and cream cheese until melted and smooth. Season with salt and pepper.
Lightly brush each corn tortilla with vegetable oil and warm them in a dry skillet over medium heat for 15-20 seconds per side until pliable. Stack on a plate.
Spoon about 1/4 cup of the chicken mixture down the center of each tortilla, roll tightly, and place seam-side down in the prepared baking dish. Repeat with all tortillas, fitting them snugly.
Pour the white sauce evenly over the rolled enchiladas, using a spatula to spread it all the way to the edges. Sprinkle the remaining 1/2 cup of Monterey Jack cheese over the top.
Bake uncovered for 20-25 minutes, until the sauce is bubbly and the cheese is melted and lightly golden. If you like a browned top, broil for an additional 1-2 minutes.