Rinse chicken liver and neck; pat dry. Using kitchen shears, remove backbone from chicken, and transfer to a medium stockpot. Add neck. Cut off wings, legs, and thighs; add to pot. (Reserve liver for Chicken Roulade.)
Remove skin from breast. Cut breast off bone, and add bone to pot. (Reserve breast for Chicken Roulade.)
Add 10 cups water, parsley sprigs, carrot,
garlic, peppercorns, and 1 teaspoon salt to pot. Bring to a boil over medium-high heat. Reduce to a simmer, and cook 45 minutes, skimming foam from top.
Using tongs, transfer the wings, legs, thighs, and carrot to a plate, and let cool slightly. Remove meat from bones, and finely chop. Transfer 1/2 cup meat to a small bowl; reserve remainder for another use. Return bones to pot. Cut 8 thin slices from carrot; trim into heart shapes, and set aside.
Simmer stock 45 minutes more. Pour through a sieve lined with cheesecloth into a bowl; discard solids. Set aside, reserving 1/4 cup for Chicken Roulade.
Stir cheese, chopped parsley, 1/4 teaspoon
salt, the pepper, and egg in bowl with chicken; cover, and set aside.
Roll out 2 wonton wrappers on a lightly floured surface to 4 1/2 inch squares. Place a tablespoon of chicken filling in center of a wrapper. Place 2 carrot pieces on top of filling. Moisten edges of wrapper
with a wet pastry brush. Top with other wrapper. Press edges with fork tines to seal. Repeat to make 4 ravioli.
Bring chicken stock just to a simmer in a medium saucepan. Meanwhile, bring a medium pot of water to a boil; add 1 tablespoon salt. Add the ravioli, 1 at a time, and cook 2 minutes. Using a slotted spoon, transfer ravioli to the stock. Divide stock and ravioli evenly between 2 serving bowls.