Shred chicken, discarding skin and reserving bones. In a medium pot, bring chicken bones and 9 cups water to a boil over high. Reduce heat and simmer, partially covered, 1 1/2 hours. Strain stock through a fine-mesh sieve into a heatproof container; let cool to room temperature. Cover and refrigerate for use in Baked Polenta with Sausage and Artichoke Hearts and Tortellini Soup with Peas and Spinach.
Meanwhile, stir together apple, cucumbers, scallions, and mayonnaise. In a large skillet, heat chicken, mustard, vinegar, honey, and 1/4 cup water over medium-high, tossing to combine, until warmed through, 3 to 5 minutes; season with salt and pepper. Divide chicken mixture among buns and top with apple-cucumber slaw.