Step 1 Make chicken stock. Place the chicken, carrot, celery, onion, peppercorns, parsley, thyme and bay leaf into a large pot. Add 10 cups of water. Bring to a boil, then turn down the heat to a low simmer (the water should be barely bubbling). Simmer uncovered for 2 to 2 1/2 hours, occasionally skimming off the fat and impurities that rise to the surface.
Step 2 Carefully remove the chicken to a bowl. Strain the stock through a fine-mesh strainer. Discard the solids.
Step 3When the chicken is cool enough to handle, remove the skin and the bones and discard. Shred the dark meat and set aside. Reserve the white meat for another use.
Step 4In a small skillet, heat the olive oil and quickly saute the chicken livers until pink inside, about 2 to 3 minutes.
Step 5Place the strained stock in a large saucepan. Bring it to a boil and turn down to a simmer. Add the basil sprig, chopped tomato and fresh tagliatelle. Cook until the pasta is done, 2 to 3 minutes. Stir in the livers and shredded chicken.
Step 6Season to taste with salt and pepper. Serve in warmed, shallow soup bowls. Garnish with parsley leaves.