Preheat the oven to 375°F.
Carefully remove the skin from the roasted chicken and spread it out on a baking sheet lined with parchment paper. Place the chicken skin in the oven and cook for about 10 minutes, until it gets crispy. Remove from the oven, and when it cools, crumble it into small pieces and place in a small bowl.
While the skin is in the oven, pull the meat from the chicken bones and shred it using a fork. Place the meat in a bowl and discard the bones. Reserve the juices.
Switch your oven to broil. Spread the chicken meat out on a baking sheet, and drizzle on the olive oil. Place meat about 4 inches from the broiler and cook for about 2 minutes, until it gets a browned and crispy edge. Stir it around and broil for another 2 minutes. Remove it from the broiler and set aside.
Melt the butter in a large, heavy pot over medium-high heat. Add the carrots, shallots, and 1 teaspoon of salt, and sauté for about 6 minutes, until softened. Add a few grinds of black pepper and the Aleppo, and cook for 1 minute more. Add the sage and pine nuts, squeeze the lemons into the pot, then throw the lemon quarters in, too. Pour the chicken into the pot and toss. Decrease the heat to low and let it simmer while the pasta is cooking.
Fill a 6-quart pot with water. Bring it to a boil and add 2 tablespoons of salt. Add the radiatore and stir. Cook until slightly firmer than al dente, 1 minute less than it says on the package, and drain, reserving ½ cup of cooking water.
Pour the pasta into the pot with the chicken. Add the reserved chicken juices and the panko, and toss. Taste for salt, adding more if needed. Add a little of the reserved cooking water if it looks too dry. Top with the crispy chicken skin and a drizzle of olive oil, and serve family-style.