Char the onions and garlic in a large stockpot over medium high heat for 2-3 minutes, until well-colored, but not burnt. Add the beef bones, chicken carcasses and water, and bring to a boil. Skim off the foam, gray crud and fat as they rise to the surface. Boil for 45 minutes to an hour, or until there isn't much foam being produced any longer.
Add carrots, star anise, cinnamon, ginger, chiles and salt. Lower heat so the broth gurgles a few times each second. Let the broth cook at this low heat overnight, or for at least 8 hours.
Skim broth one more time, and strain through a fine mesh strainer (a Chinois or “China cap”) or a colander lined with cheesecloth. Stir in fish sauce. Rinse out the pot, return the broth to the pot, and bring back to a boil before serving.
To serve, divide vermicelli, chicken breast, cilantro and green onion between 8 large bowls. Ladle broth into each, and finish with a squeeze of lime.