Preheat the oven to 425 degrees F. Lightly oil a 9×13 baking dish or coat with nonstick spray.

Melt the butter in a large saucepan over medium heat. Whisk in flour until lightly browned, about 1 minute. Gradually whisk in chicken broth and cook, whisking constantly, until slightly thickened, about 1-2 minutes. Stir in sour cream and green chiles.

In a medium bowl, combine chicken and 1 cup Monterey Jack cheese.

To assemble the enchiladas, lay tortilla on a flat surface and spoon 1/4 cup of the chicken mixture in the center. Roll the tortilla and place seam side down onto prepared baking dish. Repeat with remaining tortillas and chicken mixture. Top with sour cream sauce and remaining Monterey Jack cheese.

Place into oven and bake until lightly golden and bubbly, about 20-25 minutes.

Serve immediately, garnished with cilantro, if desired.