Preheat the oven to 375 degrees Fahrenheit.
Melt the butter with the olive oil in a medium saucepan over medium heat. Once the butter has melted, add the flour. Whisk continually for about a minute or until fragrant and golden.
Slowly add the chicken broth and whisk continuously until gently simmering and starting to thicken, about 5 minutes.
In a medium bowl, combine the (room temperature) sour cream with a ladle of the chicken broth mixture.
Transfer the sour cream and chicken broth mixture back into the saucepan (containing the chicken broth) and add the can of diced green chilies. Return to a simmer, and whisking often, allow the mixture to thicken for about 5 minutes.
Remove from heat and stir in one cup of cheese until fully melted.
Spread a ladle of the sour cream sauce in the bottom of a 9x13-inch casserole dish. Reserve and set aside 2 cups of the sour cream sauce for later.
Stir the shredded chicken into the remaining sauce and set aside.
Warm the tortillas in the microwave for 30 seconds then flip the stack over and heat for another 15 seconds.
Set up an assembly line with the warm tortillas in the middle, the sauce-covered chicken and shredded cheese on one side, and the casserole dish on the other.
To assemble, spoon two to three tablespoons of the chicken into the center of the tortilla and sprinkle with cheese on top. (Use approx. one cup of cheese for assembly of the enchiladas and reserve the remaining one cup of cheese for topping)
Roll up the tortilla and place it in the pan, seam side down. Repeat the process until you have 9-10 enchiladas and most if not all the chicken has been used.
Drizzle the reserved sour cream sauce on top of the enchiladas then sprinkle the remaining cheese on top.
Bake for 20-25 minutes or until the edges are golden and bubbly.
Let them cool for 5 minutes before serving with sour cream, Pico de Gallo, jalapeño slices, freshly minced cilantro, and lime wedges if desired.