Preheat oven to 350 degrees. In small saucepan over medium-high heat, melt butter until foaming. Add flour and cook 1 minute, stirring constantly.
Stir in enchilada sauce, chicken broth, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Bring to boil; reduce heat and simmer uncovered until thickened, 10 to 15 minutes. Keep warm over low heat.
In medium saucepan or skillet, heat oil until shimmering. Add onions and green chiles. Heat until onions have softened, about 5 to 7 minutes.
Add chicken and sour cream. Stir until creamy, about 3 minutes. Keep warm over low heat.
On microwave safe plate, wrap corn tortillas in damp paper towels. Microwave 40 seconds.
Spread 1/2 cup enchilada sauce in bottom of 8 1/2-inch by 11-inch baking dish. Working with 1 tortilla at a time, add 1/4 cup of creamy chicken filling to a warmed tortilla.
Roll tortilla around filling and place seam-side down in baking dish. Repeat with remaining tortillas.
Pour remaining enchilada sauce over rolled enchiladas and sprinkle with Monterey jack. Bake until cheese is hot and bubbly, about 25 to 30 minutes. Remove from oven and let stand 5 minutes. Garnish with scallions and serve.