Put the chicken into a large soup pot. Add enough water to cover completely, then add onion, bay leaves, butter, salt, and pepper. Cover the pot, set over high heat, and bring to a boil. Reduce heat to low, partially cover, and very gently simmer until the chicken is cooked through, about 1 hour.

When the chicken is done, transfer it to a cutting board, leaving the broth and bay leaves in the pot. When chicken is cool enough to handle, pull the meat from the bone in small pieces and set aside (discard bones, skin, and other waste).

Put the flour, baking powder, and salt in a large bowl and whisk to blend. Add the oil and stir to coat the flour, then add the water and beaten eggs. Stir just enough to combine, then knead with clean hands until evenly mixed.

Turn the dough out onto a generously floured board and divide in half. Take the first portion and roll out into a thin rectangle. Slice the dough lengthwise into 1/2-inch-wide strips, then cut cross-wise into pieces 4 inches long. Repeat with remaining dough.

Bring the broth back to a boil. Drop the dumpling strips into the boiling liquid. Reduce heat to a simmer and cook uncovered, stirring occasionally to prevent sticking, until the dumplings are tender (they’ll puff up a bit), about 20 minutes. Return the chicken to the pot in the last 5 minutes of cooking. Taste and add additional salt and pepper as needed. Serve hot, garnished with parsley.