Preheat the oven to 350°F (175°C). In a small saucepan set over medium heat, allow the butter to melt until foamy. Whisk in the flour and stir constantly for about 1 minute to create a roux.
Pour in the enchilada sauce and chicken broth. Season lightly with salt and black pepper (¼ teaspoon each recommended). Allow the mixture to come to a boil, then lower the heat and simmer uncovered for 10 to 15 minutes, or until the sauce thickens. Keep warm on the lowest setting.
In a separate skillet or medium saucepan, warm the oil until it shimmers. Add in the diced onions and drained green chilies, sautéing for about 5 to 7 minutes until soft. Mix in 1 cup of the shredded chicken and all the sour cream, stirring until well combined and warmed through—approximately 3 minutes.
While the filling heats, wrap the stack of corn tortillas in a damp paper towel and microwave them for 30 to 40 seconds to soften them for rolling.
Coat the bottom of a 9x13-inch baking dish with ½ cup of the prepared enchilada sauce. Working one at a time, place about ¼ cup of the chicken mixture in the center of each warm tortilla, roll tightly, and arrange seam-side down in the baking dish.
Once all the tortillas are filled and arranged, pour the remaining enchilada sauce evenly over the top. Sprinkle with the shredded Monterey Jack cheese.
Bake uncovered for 25 to 30 minutes, until the cheese is melted and bubbling. Remove from the oven and let rest for 5 minutes before garnishing with sliced scallions and serving.