To skip the whole boiling-a-chicken-for-broth situation, substitute 8 cups prepared chicken broth and 2 cups leftover chicken. Start from instructions section "To make the soup."
In a large pot over medium-high heat, add chicken, cold water, and salt. Bring to a boil, reduce heat to low, and skim the foam off the top of the water as it rises.
When the foam has subsided, add onion and garlic to the pot. Simmer gently (bubbles should barely break the surface at irregular intervals) until the chicken reaches 165 degrees when tested with a thermometer, about 1 hour.
Remove chicken from pot to a rimmed baking sheet or large bowl. Separate chicken, discarding skin and bones, and shred or chop into 1/2-inch pieces. You should have at least 2 cups chicken.
Strain the broth through a fine-mesh strainer or cheesecloth and reserve. You should have at least 10 cups broth.
To make the soup:
In a large pot or Dutch oven, heat butter over medium-high heat until foaming. Add onion, carrots, and celery and cook until softened, about 5 minutes. Stir in garlic until fragrant, about 30 seconds. Stir in the reserved broth and reduce heat to low.
In a medium bowl, beat the eggs with a fork, then gradually add the flour in batches until you have a smooth paste. Place a colander over the soup pot, then add the dumpling paste to the colander.
Using a spatula or your hand, press the paste through the holes into the soup. Add reserved chicken and simmer until the dumplings are cooked through, about 10 to 15 minutes. Season to taste with salt and pepper (I like 1/2 teaspoon salt and 1/4 teaspoon pepper) and garnish with dried parsley.
Notes
Cold water: Always start with cold water. This helps keep the broth clear, not cloudy. The amount of water used and the length of simmering time will determine the intensity of the broth.
Onion: Use one onion total for both the broth and the soup. Peel and chop the onion, then reserve about 1/4 cup for the broth. Use the rest in the soup.