For the soup: Place the chicken pieces and the backbones in a large pot and cover with 16 cups water. Bring to a boil, then reduce the heat to a low simmer and poach the chicken until cooked through, 25 to 30 minutes.
Use tongs to remove the chicken pieces and set aside to cool; discard the backbones. Once the chicken cools enough to handle, pull the meat from the bones, shred and roughly chop; set aside. Strain the chicken broth and keep warm.
In a medium pot, melt the butter over medium heat, and stir in the carrots, garlic, celery, leeks, onions and poultry seasoning. Cook the vegetables until they begin to soften, 8 to 10 minutes, then stir in the flour.
Add the reserved chicken broth, bring to a boil, then reduce to a simmer and cook until the soup thickens slightly and vegetables soften, about 25 minutes. Stir in the shredded chicken and bring the soup back to a low simmer.
For the dumplings: In a medium bowl, add the flour. Use two butter knives to cut in the milk, butter, baking powder, salt and eggs. Add additional milk, 1 tablespoon at a time, if necessary. The dough should be moist but not too wet.
Pull the dough into 1/4-cup pieces and place directly in the simmering soup; cover and cook for 10 to 12 minutes. Sprinkle with the chopped parsley and serve.