Grease the slow cooker crock with the oil.
Stir enchilada sauce, broth, onion, garlic, green chiles, chili powder, cumin, paprika, salt, and pepper in the cooker.
Nestle the chicken into the sauce. Cover and cook on Low 4½–5 hours or High 2½–3 hours, until tender.
Transfer chicken to a board. Reserve 1 cup of the sauce in the cooker. Shred chicken and stir in lime juice.
Warm tortillas in a damp towel in the microwave until pliable, about 30–45 seconds.
Fill each tortilla with shredded chicken and a little cheese. Roll and place seam-side down into the saucy cooker.
Pour the reserved sauce over the rolls and sprinkle on the remaining cheese.
Cover and cook on High 30–40 minutes, until bubbly and melted. Rest 10 minutes, then garnish with cilantro and serve with sour cream.