Pat both the short ribs and chicken dry, then season them generously with salt and pepper. In a large Dutch oven or stockpot over medium-high heat, warm the olive oil until it shimmers. Brown the beef on all sides, about 10 minutes total, then remove and set aside. Repeat the process with the chicken, cooking until both sides are well-browned. Remove the chicken, discard the skin once cool enough, and reserve.
Without draining the rendered fat, sauté the onions and celery in the same pot over medium-high heat until softened, about 3 to 5 minutes. Deglaze the pot with chicken broth, scraping up any browned bits. Add bay leaves, then return the beef and chicken to the pot. Bring to a boil, then reduce heat, cover, and simmer until the chicken reaches an internal temperature of 175°F, about 25 to 30 minutes.
Take out the chicken, and once cooled, discard the bones and shred the meat. Refrigerate the shredded chicken. Continue simmering the broth with the beef until the meat becomes tender, around 75 to 90 minutes more. Remove the beef, discard bones, fat, and inedible tissues, then shred.
Strain the broth through a fine-mesh sieve, discarding solids. Allow the liquid to settle, skim off the fat, and return the defatted broth to the pot.
Introduce the cabbage, diced tomatoes, rutabaga, carrots, and potatoes to the broth. Let the stew simmer over medium heat for 30 to 35 minutes until all vegetables are tender. Stir in the shredded chicken and peas, cooking just until heated through. Adjust seasoning with salt and pepper as needed. Offer lemon wedges on the side, if desired.