Before starting the preparation, preheat your oven to 400 degrees. Take the roasted chiles and peel them. Then take out their seeds and cut the chiles.
Take a frying pan and turn the stove to medium heat; put in the butter and oil. When the butter has melted, add the garlic in the pan. Let it cook until you can smell the fragrance, this will take about 30 seconds. Now sprinkle the salt, pepper and add the chiles. Cook them for around three minutes and stir occasionally in between. Now it is time to put in the chicken broth. Let the sauce simmer until it gets thick and then reduces by one third. This will take around 10 to 12 minutes.
Start preparing the tortillas side by side. Take a frying pan, small in size. Put in the remaining chicken broth and let it simmer on medium heat. Take one tortilla at a time and dip it for a little time in the broth. This will make them soft. Put all the tortillas one by one in a baking sheet. You will need more than one sheet as the tortillas should not overlap.
Take 1 ¼ cups of the shredded cheese and divide it over the tortillas equally. Then put the shredded chicken on each. Wrap the tortilla in a roll shape around the filling. Place them with the seam side on the sheet. Top the tortillas with the green Chile sauce. Then sprinkle the remaining ¾ cup of cheese on it. Bake it for around 15 to 20 minutes; check if the cheese is bubbling. Serve it with sour cream on the side.