In a deep skillet, bring the chicken broth and chicken breasts to a simmer over medium heat. Reduce the heat slightly and poach the chicken, turning occasionally, until just cooked through, about 10 minutes. Using a slotted spoon, transfer the chicken to a plate and let cool.
Crumble the 4 handfuls of tortilla chips into a food processor and add 3 cups of the hot broth; let sit to soften. Reserve the remaining 1 cup broth.
In a large saucepan, heat the oil over medium heat and add the scallions, oregano and cumin; cook until the scallions are softened, about 2 minutes. Pour the remaining cup of broth into the scallion mixture, then stir in the beans and salsa and bring to a simmer.
Puree the chips and broth in the food processor until smooth, then whisk the chip mixture into the bean mixture. Shred the chicken, add to the chili and cook over medium-low heat, stirring, until heated through. Serve with extra tortilla chips.