To Poach The Chicken: Place whole chicken in a large heavy bottomed saucepan. Add in carrots, onion, salt, peppercorns, and bay leaves. Pour in enough water to cover chicken. Bring pot to a boil, reduce heat and cover, and simmer for about an hour (skimming fat off top every once in a while, if necessary), until chicken is fully cooked. Remove chicken from pot and let cool.
While chicken is cooling, bring liquid back to a boil and cook down for about 15 minutes. Remove from heat and reserve 1 cup of broth to make the sauce. (You can strain the remaining broth and freeze it for another recipe).
Once chicken is cool, use two forks to shred it into ~1 inch pieces, removing and discarding all bones.
To Make the Sauce: In a small pan, melt butter. Cook down onion until translucent. Remove from heat.
In a food processor, add onion and walnuts and ½ cup of reserved chicken broth. Puree until smooth. Add in salt, paprika, cayenne pepper, and torn bread pieces. Puree until mixture has the consistency of hummus. Add additional reserved broth if needed.
Pour walnut sauce over shredded chicken and mix well to combine. Adjust salt to taste, and then top with additional paprika before serving.