Pat beef and chicken dry with paper towels and season on both sides with salt and pepper.
In a large Dutch oven or large stockpot (at least 5 1/2 quarts), heat olive oil until just smoking. Add beef and cook, flipping occasionally, until browned on all sides, about 10 minutes total. Remove from pot and set aside.
Add the chicken to the pot and cook, flipping occasionally, until browned on both sides, about 10 minutes total. Remove from pot set aside. When the chicken is cool enough to handle, remove and discard skin. Do not drain fat from pot.
In the same pot, reheat rendered fat over medium-high heat until shimmering. Add onions and celery, and cook until softened, about 3 to 5 minutes.
Stir in broth and bay leaves, scraping up any browned bits on the bottom of the pot. Add back short ribs and chicken, and bring to boil. Reduce heat to low, cover, and simmer until chicken is 175 degrees on an instant-read thermometer, about 25 to 30 minutes.
Remove chicken from pot. When chicken is cool enough to handle, remove and discard bones. Shred chicken into bite-sized pieces. Cover and refrigerate chicken.
Continue cooking the stew until the beef is tender, about 75 to 90 minutes longer. Remove beef from pot. When beef is cool enough to handle, remove and discard fat, bones, and any inedible connective tissue.
Strain broth through a fine-mesh strainer, discarding solids. Allow liquid to settle, about 5 minutes, then skim off fat and return liquid to pot (expect 1-2 cups of fat; see note 3).