Preheat the oven to 375°F and grease a 9x13-inch baking dish; set aside. Heat 1 Tbsp oil in a skillet over medium-high heat. Heat the tortillas one at a time, until softened, about 5 seconds on each side. Transfer to a paper towel-lined plate and cover with foil to keep warm. Add more oil as needed to heat remaining tortillas.
In a small saucepan over medium heat, melt the butter. Whisk in the flour until mixture is bubbling, then whisk in chicken stock, sour cream, canned green chiles, ½ fresh lime juiced, 1 tsp salt, and ¼ tsp ground black pepper. Remove from heat.
In a large bowl, stir together the shredded chicken, ¾ cup green chile sauce, 2 ½ cups cheese, onion, and diced red chile. Divide filling among the tortillas, roll up each tortilla, and transfer seam side down to the baking dish. Arrange the rolled tortillas in two rows of 7 across the width and the remaining 2 in the middle; set aside. Pour remaining sauce evenly over the enchiladas. Sprinkle on remaining Monterey Jack cheese.
Bake until sauce is bubbly and heated through, about 35-40 minutes. Sprinkle with a little paprika (optional), green onions, chopped cilantro, and sliced red chile peppers. Serve with Cotija cheese and lime wedges.