Put leek greens, carrot peels, celery leaves, garlic, parsley stems, thyme, bay leaf and star anise at bottom of a large Dutch oven or soup pot.
Place chicken on top of the vegetables.
Add remaining 1 teaspoon salt and the peppercorns; pour in 8 cups cold water, or enough to just submerge the chicken.
Bring to a boil over high heat; reduce heat to a simmer and partly cover the pot; cook until chicken is cooked through and very tender; 45 to 55 minutes.
While the chicken cooks, thickly slice 3 of the leek whites into ½-inch coins.
Thinly slice the remaining leek white; set thinly and thickly sliced leeks aside separately.
When the chicken is tender, using tongs, transfer to a large bowl or plate to cool; strain the broth into a large bowl, discarding vegetables and herbs.
Wipe out the Dutch oven and return it to medium-high heat; add butter, letting it melt.
Add the thick leek coins (save the thinly sliced leek for garnish), a pinch of salt and pepper; sauté until tender and golden brown; about 5 minutes.
Transfer leek whites to a bowl (they can go on top of the chicken if there’s room).
Pour broth back into the pot; bring broth to a brisk simmer and stir in barley, carrots and celery; let broth simmer, uncovered, until the liquid is reduced by a third and the barley and vegetables are tender; 40 to 50 minutes.
While the broth is simmering, use a fork or fingers to shred chicken into bite-size pieces, discarding skin and bones.
Once the barley and vegetables are tender, stir in shredded chicken, sautéed leek whites and the prunes; simmer for another 5 minutes to allow the prunes to soften (some might disintegrate); taste and add salt, if required
To serve, garnish with thinly sliced leek whites and parsley leaves.