Thaw and clean the whole roasting chicken and remove the giblets. Place the chicken in a large stockpot and cover it with water.
Bring it to a boil and let it cook for about an hour.
Remove the chicken and put it in a large bowl to cool.
Strain broth to remove loose particles and return it to the stockpot. Keep the broth on a low heat.
In a medium-sized bowl, mix two cups of chicken broth with two eggs. Add flour until it becomes a thick paste (should take about 2 to 3 cups).
Generously sprinkle flour on the counter. Take a handful of paste and work more flour into the mixture while forming a doughy ball. (Make sure it is firm enough to roll out without sticking to the work surface and rolling pin.) Roll the dough ball flat with a rolling pin.
Use a knife or pizza cutter to cut dough into strips that are one-inch wide and two inches long.
Drop the dumplings into the stockpot with the chicken broth. Repeat until you are out of dough.
Once the chicken is cool, remove the skin, pull it apart, cut into small pieces, and discard the bones. Add the chicken pieces to the stockpot (after adding the dumplings) and return to a boil. Stir.
Cook the chicken and dumplings for another 10 to 15 minutes.
Add salt and pepper to taste. Add more chicken broth, chicken bouillon cubes, and butter to add flavor, stir occasionally until you reach the desired consistency.