Preheat the oven to 350°F (175°C). Lightly grease a 9×13 baking dish.
If using corn tortillas, flash-fry them: add enough canola or vegetable oil to coat the bottom of a skillet and heat over medium-high. Fry each tortilla about 10 seconds per side until soft with a few bubbles but not browned; drain on paper towels. (This step is optional; see notes for microwave option.)
Make the green chile sour cream sauce: in a large saucepan, melt 2 tablespoons butter with 2 tablespoons olive oil over medium heat. Add 1/2 cup chopped onion and sauté about 3 minutes until softened.
Sprinkle in 1/4 cup flour and add the minced garlic; cook 2 minutes, stirring, until the mixture is thick and fragrant.
Reduce heat to low and slowly whisk in 2 cups low-sodium chicken broth. Then whisk in 1 cup salsa verde, the 14 oz can diced green chilies (with juices), 1 teaspoon ground cumin, 1 teaspoon chicken bouillon, 1/2 teaspoon salt, and 1/4 teaspoon cayenne. Bring to a boil, then simmer until slightly thickened.
Remove the sauce from the heat and stir in 1 cup sour cream until smooth. Reserve about 1/2 cup of the sauce for the bottom of the baking dish.
Prepare the filling: in a large bowl, combine 4 cups shredded chicken with 1 cup of the green chile sour cream sauce and 2 cups of the shredded Monterrey Jack cheese. Mix until evenly coated.
Spread the reserved 1/2 cup sauce in a very thin layer over the bottom of the prepared 9×13 baking dish.
Assemble the enchiladas: place about 1/2 cup filling in the center of each tortilla, roll tightly, and arrange seam-side down in the baking dish. Repeat with remaining tortillas and filling.
Pour the remaining green chile sour cream sauce evenly over the rolled enchiladas, then sprinkle the remaining 1 cup shredded Monterrey Jack cheese on top.
Bake uncovered at 350°F for 30–35 minutes, until the cheese is fully melted and the center is hot. If desired, broil briefly until the cheese is golden—watch carefully to avoid burning.
Serve hot with additional sour cream, chopped tomatoes, sliced avocados, cilantro, lime, chips, and hot sauce as desired.