How to Make It
Preheat the oven to 400°. Put the whole chicken, quartered onion, carrot halves and peppercorns in a large soup pot and add enough water to cover the chicken. Bring to a boil, skimming off any scum with a spoon. Simmer until the chicken is cooked, about 45 minutes.
Carefully remove the chicken from the pot and pull the meat from the bones; discard the skin. Return the bones to the pot and simmer for 30 minutes longer.
Meanwhile, put the beet wedges in a medium baking dish. Add 2 tablespoons of the olive oil and 1/4 cup of water and season with salt and pepper. Cover with foil and roast for about 1 hour, until the beets are very tender. Let cool. Stir the vinegar and 2 tablespoons of the olive oil into the pan juices.
Shred the chicken and transfer it to a bowl. Strain the chicken broth from the soup pot. Pour 1 cup into a bowl and save the rest for another use. Add the bread to the broth and let stand until softened. In a blender or food processor, pulse the walnuts with the garlic until finely chopped. Add the soaked bread and puree until creamy and smooth. If necessary, add a few extra tablespoons of broth until the sauce is the consistency of thin sour cream. Season with salt and pepper. Stir the sauce into the chicken and season with salt and pepper.
In a small skillet, heat the remaining 2 tablespoons of olive oil with the Aleppo pepper over moderate heat until fragrant, about 1 minute. Drizzle the spicy oil over the chicken salad.
Arrange the arugula on a platter and top with the beets and their dressing. Spoon the chicken salad alongside and serve with the pita.