In a medium saucepan, combine the salsa verde and chicken broth. Nestle the chicken breasts into the salsa mixture. Bring to a boil, then reduce to a gentle simmer and cook, covered, for 12-15 minutes, or until the chicken is just cooked through (if a lot of foam rises to the surface, skim it off and discard).
Transfer the chicken to a cutting board or plate and let cool. Simmer the sauce, uncovered, for 5 minutes. Once cool enough to handle, shred the chicken and stir it back into the salsa verde sauce. Remove the pot from the heat and keep warm.
In the meantime, heat the olive oil in a small skillet over medium heat. Add the diced onion, and cook, stirring occasionally, until softened and lightly browned, about 5 minutes. Add the garlic, cumin and oregano. Cook, stirring, until fragrant, about 30-60 seconds longer.
Scrape the onion mixture into the pot with the chicken. Do Ahead: The chicken in its sauce can be refrigerated for up to 3 days. Reheat before proceeding.
When you’re ready to serve, reheat the chicken and sauce. Add the tortilla chips, and toss gently in the sauce (alternatively, for a less saucy version, place the chips in a skillet and add just enough chicken and sauce to lightly coat; cook over medium heat until warmed through).
Divide the chilaquiles between shallow bowls (it will thicken as it sits). Add the toppings of your choice, and devour!