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Preheat your oven to 375°F. Lightly grease a 9x13 inch baking dish with olive oil. Set it aside while you prepare the filling and creamy sauce.
In a large bowl, mix shredded chicken, garlic powder, onion powder, salt and black pepper. Add one cup Monterey Jack cheese. Stir well until everything is evenly combined.
Lay one tortilla flat on a clean surface. Add a scoop of chicken mixture in the center. Roll it tightly and place seam side down in the baking dish. Repeat with remaining tortillas.
In a saucepan, melt butter over medium heat. Add flour and whisk for one minute. Slowly pour chicken broth while whisking. Cook until smooth and slightly thick.
Turn heat to low. Stir in sour cream and green chilies. Mix gently until smooth and creamy. Do not boil or the sauce may separate.
Carefully pour the warm sauce over the rolled tortillas. Make sure all enchiladas are covered. Spread evenly with a spoon for full flavor in every bite.
Sprinkle remaining Monterey Jack and cheddar cheese over the sauce. Cover the dish completely so it melts into a golden, bubbly topping during baking.
Place the baking dish in the oven. Bake for 20 to 25 minutes. The cheese should be melted and lightly golden. Let rest five minutes before serving.