To make the stuffing: In a small skillet, heat oil over medium heat; add livers and cook, stirring, 1 to 2 minutes. Place in a large bowl along with breadcrumbs, ham, garlic, parsley, sage, red pepper, and allspice; mix well. Season with salt and pepper; stir in egg until well combined.
Spoon stuffing into the cavity of the chicken; truss to enclose. Place in a large stockpot; add 6 quarts water and 1/2 teaspoon salt. Press cloves into onion and place in a piece of cheesecloth along with bay leaves, thyme, parsley, peppercorns, celery, and cinnamon stick; tie cheesecloth with kitchen twine to enclose and add to stockpot. Cover and bring to a boil over medium-high heat, skimming often. Reduce heat to medium, uncover, and cook, simmering, until broth gradually reduces, about 30 minutes.
Tie leek pieces in a bundle with kitchen twine and add them to the stockpot along with the carrots and the turnips. Add more water, if necessary, to just cover chicken and vegetables; season with salt. Continue simmering until chicken and vegetables are very tender, the chicken leg meat pulls away from the bones, and juices run clear when chicken thigh is pierced with a fork, about 1 hour.
Transfer chicken to a cutting board; loosely cover to keep warm. Transfer vegetables to a large bowl using a slotted spoon; cut twine from leeks. Return pot to stove over medium-high heat and bring to a boil. Cook until liquid has reduced by a third (you should have about 2 1/2 quarts), 15 to 25 minutes. Strain liquid into a medium saucepan, discard cheesecloth packet, and season with salt. Add pasta and place saucepan over medium-high heat; bring to a boil, and reduce heat to a simmer. Simmer until just tender, about 2 minutes. Garnish with parsley.
Remove kitchen twine from chicken. Remove stuffing from cavity and transfer to a large serving platter. Place vegetables around stuffing and set chicken on top. Garnish with sage and serve with pasta and sea salt.