How to Make It
In a large saucepan, combine the chicken legs with the large chunks of carrot and celery, half of the chopped onions and the water. Simmer over moderate heat until the chicken legs are cooked through, about 15 minutes. Transfer the chicken to a plate. Let cool slightly, then remove the meat from the bones and let cool. Strain the chicken broth and discard the vegetables.
In a medium bowl, combine the flour with 1 cup of the chicken broth, adding a little of the fat from the broth; stir to form a stiff dough. Turn the dough out onto a lightly floured board and knead until smooth. Wrap the dumpling dough in plastic wrap and let rest at room temperature for 30 minutes.
Cut the dough into quarters. Working with 1 piece at a time, roll out the dough a scant 1/4 inch thick on a very lightly floured work surface. Cut the dough into 3-inch-wide strips and transfer to a sheet of wax paper. Working on the paper, cut each strip crosswise into 1-inch-thick dumplings and transfer the paper to a baking sheet; do not separate the dumplings. Repeat with the remaining dough, stacking each layer on the baking sheet. Freeze the dumplings until solid, at least 4 hours and preferably overnight.
In a large enameled cast-iron casserole, combine the sliced carrots and celery and the remaining chopped onion with 4 cups of the broth and bring to a boil. Season well with salt and pepper and cook until the vegetables are just tender, 7 to 8 minutes. Add the cooked chicken and return to a boil.
Separate the frozen dumplings and add them to the simmering broth a few at a time, submerging them in the broth. Bring the broth to a simmer and cook until the dumplings are tender, about 15 minutes; tuck the dumplings into the broth occasionally. Spoon the chicken and dumplings into bowls and serve.