Melt butter in a large saucepan over medium heat. Whisk in flour and cook for 1 minute, stirring constantly, until it forms a smooth paste and turns light golden. This creates the foundation for your velvety white sauce.

Gradually add chicken broth to the roux, whisking constantly to prevent lumps. Continue whisking until sauce thickens and becomes smooth, about 3-4 minutes. Remove from heat and stir in sour cream, cumin, salt, and pepper until perfectly combined.

In a mixing bowl, combine shredded chicken, diced onion, green chiles, cilantro, and 1 cup of the cheese mixture. Mix thoroughly to create a well-distributed filling that delivers flavor in every bite.

Preheat oven to 350°F. Spread a thin layer of white sauce in bottom of greased 9x13 baking dish. Place about ⅓ cup chicken mixture down center of each tortilla, roll tightly, and arrange seam-side down in prepared dish.

Pour remaining white sauce evenly over rolled enchiladas, ensuring they're completely covered. Sprinkle remaining Monterey Jack and cheddar cheese over the top for that golden, bubbly finish.

Bake uncovered at 350°F for 25-30 minutes until sauce is bubbling around edges and cheese is melted and golden brown. Remove from oven and let rest for 5 minutes before serving to allow sauce to set properly.