Heat oil over medium heat in a large Dutch oven for 1 minute. Add chicken, skin-side down, and brown for about 5 minutes per side. Remove from pan and add celery, carrot, onion, garlic and paprika. Cook for 5 minutes, or until onion is translucent. Add broth and water, then return the chicken to the pan. Pepper generously. Simmer, covered, for 45 minutes.
Meanwhile, warm the milk and saffron threads for one minute on stovetop over medium heat or in microwave. The milk will turn yellow. In a food processor, or with your fingers, quickly combine flour, baking powder, and shortening. Remove blade from bowl and gently mix in saffron milk. Gather dough into a ball and flatten it out into a 1-inch thick disk. Cover with plastic wrap and refrigerate until just 20 minutes before use.
After dough has chilled, place it on a lightly floured surface and cut into 12 rectangular pieces with a sharp knife. Just 15 minutes before serving, slip dumplings into the simmering soup. Cover and cook just below a boil until dumplings puff up and float to top. Carefully ladle some soup with 2 thighs and 3 dumplings into each bowl. Garnish with a little chopped parsley.