Set your oven to heat at 375°F (190°C).
In a mixing bowl, toss together the shredded chicken, half a cup of Monterey Jack cheese, and half of the canned green chiles.
In a medium saucepan, melt the butter over medium heat, then whisk in the flour and stir for about 1 to 2 minutes to form a roux.
Gradually add the chicken broth while whisking continuously until the sauce thickens and simmers gently.
Remove the pan from heat and blend in the sour cream along with the remaining green chiles until fully incorporated and smooth.
Fill each flour tortilla with the chicken mixture, roll them tightly, and arrange them seam-side down in a greased baking dish.
Pour the prepared sour cream sauce evenly over all the rolled enchiladas.
Sprinkle the rest of the shredded Monterey Jack cheese on top.
Bake uncovered in the preheated oven for 20 to 25 minutes or until the cheese is bubbly and light golden brown.
If desired, garnish with diced tomatoes and chopped cilantro.
Allow the enchiladas to rest for about 5 minutes before serving.
