Pat beef and chicken dry with paper towels and season with salt and black pepper.

Heat oil in large Dutch oven over medium-high heat until just smoking. Brown beef on all sides, about 10 minutes; transfer to plate. Cook chicken until browned all over, about 10 minutes; transfer to plate. When chicken is cool enough to handle, remove the skin and discard it.

Pour off all but 1-1/2 teaspoons fat from pot. Add onions and celery and cook over medium heat until softened, about 5 minutes. Stir in broth and bay leaves, scraping up any browned bits. Add beef, beef bones, and chicken, and bring to boil.

Reduce heat to low and simmer, covered, until chicken registers 175° F, about 30 minutes. Transfer chicken to bowl. When chicken is cool enough to handle, shred into bite-size pieces, discarding bones. Cover chicken and refrigerate.

Continue to simmer stew until beef is tender, about 1-1/4 hours longer. Transfer beef to plate. When cool enough to handle, shred into bite-size pieces, discarding fat. Remove beef bones and bay leaves; discard.

Strain broth through fine-mesh strainer; discard solids. Allow liquid to settle, about 5 minutes, then skim off fat and return liquid to pot.

Add shredded beef, cabbage, tomatoes, rutabaga, 1-1/4 teaspoon salt, and 1 teaspoon pepper to liquid and bring to boil. Reduce heat to medium-low and simmer until rutabaga is translucent around edges, about 15 minutes.

Stir in potatoes and carrots and cook until vegetables are tender, about 20 minutes. Add chicken and peas, and simmer until heated through, 2 to 3 minutes. Remove from heat and stir in lemon juice. Season with salt and pepper to taste. Serve hot.