Place the barley in a saucepan, add water to cover by 2 inches, and let soak, refrigerated, for 12 hours or overnight. After soaking, drain the barley and spread out on a baking sheet to dry.
Place the chicken stock in a medium saucepan and bring to a boil.
In a wide, high-sided pot, heat the butter over medium-high heat until it foams and subsides. Add 1/3 of the apple and saute for a few minutes, until it begins to soften. Add the barley and stir for 3 minutes over high heat.
Combine the grappa and wine, and stir into the pan with the barley. Add the chicken stock, a little at a time, stirring constantly, allowing the barley to absorb the liquid to become firm and creamy, like risotto. This will take about 10 minutes.
After 10 minutes, toss in the remaining apples, mint, and grated cheese and stir 2 more minutes over high heat. Serve immediately.
In a large, heavy-bottomed saucepan, heat the oil over high heat until smoking. Add all the chicken parts and brown all over, stirring to avoid burning. Remove the chicken and reserve. Add the carrots, onions, and celery to the pot and cook until soft and browned. Return the chicken to the pot and add 3 quarts of water, the tomato paste, peppercorns, and parsley. Stir with a wooden spoon to dislodge the browned chicken and vegetables bits from the bottom of the pan. Bring almost to a boil, then reduce heat and cook at a low simmer until reduced by half, about 2 hours, occasionally skimming excess fat. Remove from heat, strain, and press on the solids with the bottom of a ladle to extract out all liquids. Stir the stock to facilitate cooling and set aside. Refrigerate stock in small containers for up to a week or freeze for up to a month.
Yield: 1 1/2 quarts