Place the chicken in the slow cooker. Sprinkle with cumin, garlic, salt, and pepper. Cover with lid and cook on low for 3-4 hours. When the chicken can be easily pulled apart, shred it and toss it in the juices in the crockpot.
Preheat the oven to 350°F. Transfer the chicken to a large bowl, add 1 cup of shredded cheese, and the green chile. Stir to combine. Scoop the chicken mixture into the tortillas, about 3/4 cup of filling fits nicely in each flour tortilla. (About 1/3 cup of filling fits easily in the smaller corn tortillas.) Roll them up and place them snugly into a baking pan. Brush with olive oil. Bake 10-15 minutes, until lightly browned and crispy.
While the burritos are in the oven, melt the butter in a large skillet over medium high heat. Add the flour and whisk until smooth. Slowly add the chicken broth and whisk until smooth. Add the cumin, salt, and pepper. Cook for 3-4 minutes, whisking and simmering until thickened.
Remove from the heat. Add the sour cream and the remaining cheese, whisking until smooth again. Pour the sauce over the crispy burritos. Enjoy!
Nothing beats freshly shredded cheese in this recipe. You'll need about 16-ounces of pepper jack cheese and it will only take a few minutes to grate it. Chicken breasts will also work for this recipe. Frozen chicken works fine, just add a couple hours to the cooking time for the chicken.