Start by heating the vegetable oil in a saucepan over low heat. Once warmed, whisk in the flour and cook for about two minutes until slightly golden. Gradually add the chicken broth, enchilada sauce, cumin, salt, and a pinch of pepper. Turn up the heat to medium-high and bring the mixture to a boil. Once it thickens, set it aside to cool slightly.
Combine the shredded chicken, green chiles, sour cream, and Monterey Jack cheese in a large mixing bowl. Sprinkle in salt and pepper. Stir everything until all the ingredients are evenly mixed.
To make assembly easier, heat your tortillas gently. Stack them between two damp paper towels and microwave for 20-30 seconds. This will make them pliable and less likely to tear when rolling.
Divide the filling evenly among the tortillas. Fold the edges inward and roll tightly. Place each burrito seam-side down in a greased 9×13-inch baking dish.
Pour the prepared sauce generously over the burritos. Cover the dish with foil and bake in a preheated 350°F oven for 20 minutes. Remove the foil, sprinkle shredded cheese on top, and bake for another 10 minutes until the cheese is melted and bubbly.
Finish with green onions and any optional toppings you love. Serve hot, straight from the oven!