Place all ingredients except cornstarch in a 4-quart saucepan; bring to a boil. Reduce to simmer and cook for 1 hour. Puree in blender and return to a boil. Mix 2 tablespoons cornstarch with 3 tablespoons water and add to sauce. As soon as sauce thickens, remove from heat and set aside.
Place all ingredients in a saucepan and cover. Bring to a boil, reduce to simmer, and cook for 40 minutes. Strain out all liquids and shred chicken using 2 forks.
To assemble enchiladas
Preheat oven to 375°. Spread one-third of verde sauce on the bottom of a large baking dish. Lay out all of the tortillas and place a pinch of cheese (1 oz.) and 3 ounces of chicken on each. Roll up tortillas and line baking pan with 2 rows of 4. Pour over remaining sauce and cover with remaining cheese. Bake for 20 minutes. Garnish with sour cream and chopped cilantro. Serves 4.