Preheat oven to 425°. Cut tortillas into quarters. Toss with olive oil until coated, then spread in a single layer on 2 baking sheets. Toast in oven about 15 minutes, turning once, until evenly crisp and golden.
Meanwhile, heat a heavy skillet over medium heat. Toast chiles just until fragrant, turning, about 30 seconds. Stem chiles (keep seeds) and add to a blender. Pour in chicken broth and let sit until chiles have softened, about 15 minutes.
Add tomatoes, garlic, and onion to chiles and blend into a smooth salsa.
In a large frying pan, simmer salsa over medium heat, stirring occasionally, until about as thick as ketchup, 10 to 12 minutes. Stir in sugar and salt. Stir in chicken, followed by toasted tortillas (in batches if necessary). Cook until tortillas are softened but still a little crunchy, 2-3 minutes more.
Divide among 4 plates and layer with toppings.