To prepare the duck, in a baking dish just large enough to hold the duck leg quarters, combine the salt, cane sugar, and thyme. Evenly and generously coat the duck pieces with the salt mixture and arrange them in a single layer in the pan, tucking them into the salt mixture. Refrigerate uncovered overnight. Preheat the oven to 250 degrees F. Rinse the duck and pat it dry. Arrange the duck pieces in a single layer in a large, heavy pot (such as an oval Dutch oven) or a heavy roasting pan with sides that are at least 4 inches deep. Melt the duck fat over low heat and pour it over the duck. The pieces must be submerged. Bake until the meat is spoon tender, but not falling off the bone, about 2 1/2 hours.
To use the duck at once, remove it from the fat and set it and the fat aside to cool to room temperature. Pick the duck meat from the bones, trying to keep the meat in large pieces. Discard the bones and skin. Use now or cover and refrigerate for up to 2 days. Transfer the fat to an airtight container and refrigerate until solid. Discard the layer of solidified cooking juices at the bottom of the container and refrigerate the remaining fat for up to 2 days or freeze for up to 3 months. (For longer storage, leave the duck on the bone and submerged in the fat. Cool to room temperature and then cover and refrigerate for up to 1 week. When ready to use, reheat in a 250 degree F oven until the fat is liquid and the duck is just warm.)
To prepare the dumplings, preheat the oven to 350 degrees F. Pierce the potatoes in several places with a fork and bake until tender, about 1 hour. Set aside until cool enough. While the potatoes are still warm, fold in the eggs and salt. Stir in 3/4 cup flour and then add the remaining flour a little at a time until the mixture resembles soft biscuit dough. The dough should be a little sticky, although when poked with a fingertip the dimple should not close up. To test the dough, bring a small saucepan of salted water to a simmer. Break off a 3/4 inch square piece of dough and drop it into the water. If the dumpling breaks apart in the water, add a little more flour to the dough. Pour the dough onto a lightly floured work surface and cut into 4 equal pieces. Line a rimmed baking sheet with parchment paper and lightly dust it with flour. Working with one piece at a time, roll the dough into a 3/4-inch-thick rope. Cut each rope crosswise into bite-size lengths, 1 to 1 1/2 inches. Roll each piece on a gnocchi board or the back of the tines of a fork to ridge them lightly, if you wish. (The ridges help the dumplings hold the delicious broth.) Spread the dumplings onto the prepared baking sheet.
Bring a large pot of water to a simmer. Salt generously. Set a wire rack inside a rimmed baking sheet and place them next to the stove. Add half of the dumplings to the boiling water and cook them until they float. Use a wire skimmer to transfer them to the wire rack. Cook the remaining dumplings. The dumplings can be used now as part of the finished dish, or they can be stored to use later.
To store them, refrigerate the dumplings on the wire rack until chilled. Toss the cold dumplings with enough oil to keep them moist (to prevent sticking). Transfer to an airtight container and refrigerate for up to 2 days. To poach the eggs, fill a large skillet with water to a depth of 2 inches. Add the vinegar and bring to a simmer. Prepare a large bowl of very cold water. Break 1 egg into a small bowl or cup and slide the egg into the simmering water. Repeat with each remaining egg, spacing them evenly in the skillet; cook them in 2 batches if necessary. Poach at a bare simmer until the whites are firm but the yolks are still runny, about 2 minutes. Use a slotted spoon to transfer the eggs to the cold water, taking care to not break the yolks. Use at once or refrigerate until ready to assemble the dish, up to 1 day. Set the skillet of water aside.
To assemble the dish, in a very large skillet, heat 1 tablespoon of the duck fat over medium-high heat until shimmering hot. Add half the dumplings and cook until they are golden brown on the bottom, about 3 minutes. Transfer to a plate. Heat the remaining 1 tablespoon of duck fat and cook the other half of the dumplings. Return all of the dumplings to the skillet. Add the duck and the chicken stock and cook over medium-high heat until the duck is warmed through, 3 to 5 minutes. Season with the pepper and taste for salt. Meanwhile, bring the water used to poach the eggs back to a bare simmer. Divide the duck, dumplings, and broth among 6 warmed shallow bowls. Working with 1 egg at a time, slip it back into the simmering water for 30 seconds. Lift each egg from the water with a slotted spoon and pat the bottom dry with a clean kitchen towel. Place 1 egg on top of each serving. Sprinkle with thyme leaves and serve at once.
Recipe courtesy of Sam Beall