In a large bowl, stir the salt with the egg and 2⁄3 cup lukewarm water until smooth. Stir in the flour until a stiff dough forms. Transfer the dough to a clean work surface and knead briefly until smooth. Wrap the dough in plastic wrap and let rest for 30 minutes.
Meanwhile, finely chop 1 onion and place in a medium bowl along with the pork, minced parsley, and garlic. Season with salt and pepper and, using your hands, mix the pork with the aromatics until the filling is evenly combined.
Unwrap the dough and place on a lightly floured work surface. Using a rolling pin, flatten the dough until 1⁄8 inch thick. Using a 3-inch round cutter, stamp out circles of dough. Place 2 teaspoons pork filling in the center of each circle, brush the edge of the circles with water, and fold the circles in half to create half-moon dumplings. Transfer the dumplings to a parchment paper-lined baking sheet and refrigerate until ready to use.
Halve the remaining onion and place in a large saucepan along with the parsley stems, chicken, peppercorns, bay leaf, and carrot and cover with 12 cups water. Bring to a boil, reduce the heat to maintain a simmer, and cook until the chicken is tender and the broth is reduced, about 1 1⁄2 hours. Transfer the chicken to a plate and let cool. Once cool, remove and discard the skin and bones and shred the meat into bite-size pieces.
Meanwhile, pour the broth through a fine strainer into a bowl and discard the solids. Pour the broth back into the saucepan and bring to a boil. Reduce the heat to maintain a simmer, add the dumplings, and cook until they are tender, about 3 minutes. Stir in the shredded chicken and season with salt. Ladle the broth, dumplings, and chicken into serving bowls and garnish with whole parsley leaves.