Melt the butter in a medium saute pan over medium heat. Add the onion and 1⁄4 teaspoon of the salt. Saute, stirring, until the onion is translucent but not browned, about 6 minutes. Turn the heat to low, add the broth and the chicken pieces, and very slowly bring to a gentle simmer, about 15 minutes.

Dumplings

While the chicken and broth heat up, make the dumplings: Using a fork, mix the flour and remaining 1 teaspoon salt in a large bowl until they’re combined. Add the beaten egg and toss it with the flour mixture until it comes together in a number of balls of varying sizes. With lightly oiled hands, gather these pieces together into a single ball and knead it against the side of the bowl, gathering any dry flour from the sides of the bowl. Turn the ball out onto a lightly floured cutting board and knead it several times, pushing the dough against the board with the palm of your hand. Divide the dough ball in half. Using a lightly floured rolling pin, roll out one of the balls on the board until it is 1⁄8 inch thick (if the pin sticks to the dough, scatter flour by 1⁄2 teaspoonfuls onto the flattened dough, spreading it out on the surface with your fingers). Cut the rolled out dough into strips about 3⁄4 inch wide and 4 inches long, and reserve. Repeat with the second ball of dough. Each half makes about 32 dumplings.

When the broth is simmering and the chicken is just opaque, gently remove the chicken pieces from the broth with tongs and set them aside. If there’s lots of protein foam floating on the broth and it’s bumming you out, pass the broth through a fine-mesh strainer, or a strainer lined with cheesecloth, and return the clarified broth to the pot. Bring the broth to a vigorous simmer, add the dumplings, and simmer until they’re cooked through but tender, about 6 minutes.

Return the chicken to the pot and heat through to serving temperature, about 1 minute. Divide the chicken and dumplings among 4 bowls, submerge a couple sprigs of parsley in each bowl, and serve immediately.

Variation

Chicken and Lemon–Black Pepper Dumplings Mix 1 teaspoon freshly ground coarse black pepper and 1⁄2 teaspoon lemon zest into the bowl of salt and flour before adding the egg.