Combine the chiles, garlic, chili powder, cumin, chicken broth, tomato sauce, salt and pepper in a medium saucepan over high heat. Bring to a boil. Reduce the heat to low and simmer for 15 minutes or until the chiles are soft Set aside until ready to use.
Preheat oven to 350 degrees F.
Heat the vegetable oil in a medium skillet over medium-high heat. Saute the chicken until cooked through, approx 7-9 minutes. Remove the chicken from the pan and place in a medium bowl. Add the onions to the same pan along with a generous pinch of salt, decrease the heat to medium-low and sweat for 4-6 minutes. Add the garlic and oregano and cook until the onions are tender, 2-3 minutes more. Add the chicken back to the pan and remove from heat.
Spray a 13x9 glass baking dish with non-stick spray. Place 1/2 cup of the sauce into the bottom of the dish. Dip 4 tortillas into the remaining sauce and lay them in the bottom of the dish. Cut one of the tortillas in half to evenly cover the bottom of the dish. Top this with half of the chicken mixture and 1 cup of the cheese. Starting with 4 more tortillas, repeat the layers, ending with the last 4 tortillas on top. Pour remaining sauce over the dish and top with the remaining 1 cup of cheese.
Cover with aluminum foil and bake in the oven on the middle rack for 30 minutes. Remove the foil and bake an additional 10 minutes, or until cheese on top is bubbly.