Place the chicken, broth, half of the lime juice, cumin and salt and pepper in a pot. Cover and simmer for 40 minutes. Shred and let cool slightly.
Preheat oven to 400 degrees F. Lightly grease a 9-inch square baking dish with the vegetable oil.
Place a layer of 3 tortillas on the bottom of the baking dish. Spread 1/3 of the chicken on top of the sauce. Repeat with another layer of tortillas, sauce and meat. Finish with layer of tortillas and pour the rest of the sauce on top. Sprinkle the cheese over top. Bake for 30 minutes, until top is brown and dish is bubbling.
Mix together avocado, onion, cilantro, the rest of the lime juice and salt and pepper.
Let the enchilada pie rest for 5 minutes before serving. Serve with salsa on the side.