Preheat the oven to 350°F (175°C).
If using corn tortillas, heat enough canola or vegetable oil in a skillet to cover the bottom over medium-high heat and quickly fry each tortilla about 10 seconds per side until soft with a few bubbles; drain on paper towels. (This step is optional; skip for flour tortillas.)
Make the green chile sour cream sauce: melt the butter with the olive oil in a large saucepan over medium heat. Add the chopped onion and sauté about 3 minutes until softened.
Stir in the flour and minced garlic and cook 1–2 minutes until fragrant and thickened. Reduce heat to low and slowly whisk in the chicken broth until smooth.
Whisk in the salsa verde, diced green chilies (with their juices), chicken bouillon, cumin, salt, and cayenne. Bring to a boil, then simmer until the sauce thickens.
Remove the sauce from the heat and stir in 1 cup sour cream until combined; set aside.
Prepare the filling: place the shredded chicken in a large bowl and stir in 1 cup of the green chile sour cream sauce and 2 cups of the shredded Monterey Jack cheese until evenly combined.
Spread about 1/2 cup of the sauce thinly over the bottom of a lightly greased 9×13 baking dish.
Place approximately 1/2 cup of the filling in the center of each tortilla, roll up tightly, and arrange seam-side down in the prepared baking dish.
Pour the remaining green chile sour cream sauce evenly over the rolled enchiladas and sprinkle the remaining 1 cup of shredded cheese on top.
Bake uncovered at 350°F (175°C) for 30–35 minutes, until the cheese is melted and the center is hot. If desired, broil briefly until the cheese is golden brown.
Serve topped with additional sour cream, chopped tomatoes, avocado slices, cilantro, a squeeze of lime, and hot sauce as desired.
