Place chicken in a stockpot and add enough water to cover. Pour in chicken broth; add onions and celery. Season with salt, pepper, and garlic salt. Bring to a boil and cook until an instant-read thermometer inserted into the center of chicken near the bone reaches 165 degrees F (74 degrees C), about 1 hour. Remove chicken to a platter and let sit until cool enough to handle.

Strain broth, discard onions and celery, and return broth to the stockpot. Remove chicken meat from the bones, chop or tear into pieces, and add to broth in the pot. Bring to a boil and add carrots.

Stir together eggs, 1/2 cup water, and salt in a medium bowl until well combined. Gradually add flour until dough is firm enough to form a ball; add more or less flour as needed. Pat dough out onto a flat plate. Using a butter knife, cut 2- to 3-inch slices of dough off the edge of the plate, allowing them to fall directly into the boiling broth.

Cook until carrots are tender. Sprinkle with parsley flakes to serve.