In a large stock pot heat 2 tablespoons olive oil over medium heat and add the garlic and diced onion. Sauté until the onion is soft and translucent, about 5 minutes.
Add the cumin, paprika and cayenne and cook, stirring constantly for 30 seconds. Add the chicken stock, diced tomatoes, with their liquid, the green chilies and 2 cups of water. Let simmer for 40 minutes, stirring occasionally.
While the broth simmers season the chicken breasts on both sides liberally with chili powder. In a sauté pan heat the remaining tablespoon of oil on medium heat.
Cook the chicken until browned on both sides, about 5 minutes. Remove the chicken from the pan and let cool.
Chop the chicken into chunks and stir into the broth. Continue to simmer for an additional 10 minutes.
Stir in the juice from two of the limes, half of the chopped cilantro and season with salt and pepper.
Slice the tortillas in half and then into 2-inch long strips. Fill a large sauté pan with enough oil to reach 1-inch up the side of the pan. Heat oil to 350°F.
When the oil is hot, add some of the tortilla strips to the pan and fry until golden brown, about 1 to 2 minutes. Work in batches so the pan is not overcrowded. Remove to a paper towel lined plate.
To serve, scoop the broth into bowls and top with shredded Monterey jack cheese, avocado, the remaining cilantro and tortilla strips. Cut the remaining lime into wedges to serve on the side.