In a large bowl, combine the flour, sugar, salt, and dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, and vanilla extract.
Mix the ingredients together, then knead the dough for 8-10 minutes until it becomes smooth and elastic.
Cover the dough and let it rise in a warm place for about 1 hour, or until doubled in size.
Meanwhile, prepare the filling: combine the mashed potato, heavy cream, cooked chicken, diced celery, onion, parsley, and ground black pepper.
On a lightly floured surface, roll out the dough to about 1 cm thickness. Use a glass or doughnut cutter to cut out circles.
Place a small amount of the knoephla soup filling in the center of each dough circle. Fold the dough over to form a half-moon shape, and seal the edges well to prevent the filling from escaping during frying.
Place the filled doughnuts on a baking sheet lined with parchment paper, and cover them with a clean kitchen towel. Let them rise for another 30 minutes.
In a large pot, heat the oil to about 340°F (170°C). Test the oil by dropping in a doughnut: if it immediately floats to the surface and gently sizzles, the oil is ready.
Fry the filled doughnuts for 2-3 minutes per side, until golden brown. Use a slotted spoon to remove them from the oil, and place them on paper towels to drain any excess oil.
Sprinkle the still-warm doughnuts with a little powdered sugar to enhance the unique sweet and savory flavor combination.
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