Preheat oven to 375 degrees F (190 degrees C). Grease a 9×13-inch baking dish.
Heat 2 tablespoons of oil in a skillet over medium-high heat. Fry tortillas (1 at a time) for 5 seconds on each side to soften and make them pliable. Add more oil to pan as needed. Drain between layers of paper towel and keep warm.
Divide chicken, 10 ounces of Monterey Jack cheese, and onion among the 12 tortillas. Roll up each tortilla and place seam-side down in the prepared pan.
Melt butter in a saucepan over medium heat. Add flour and whisk until mixture begins to boil. Slowly add broth, stirring with a whisk until thickened. Mix in the sour cream and chiles, heating thoroughly but do not boil, stirring occasionally. Pour mixture over the enchiladas.
Bake in the preheated oven until bubbly and heated through, about 20 minutes. Top with remaining Monterey Jack cheese and bake for 5 more minutes. Garnish with chopped green onions and cilantro.
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