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1
To prepare the duck, in a baking dish just large enough to hold the duck leg quarters, combine the salt, cane sugar, and thyme.
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2
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.
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3
Refrigerate uncovered overnight.
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4
Preheat the oven to 250 degrees F. Rinse the duck and pat it dry.
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5
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.
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6
Melt the duck fat over low heat and pour it over the duck.
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7
The pieces must be submerged.
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8
Bake until the meat is spoon tender, but not falling off the bone, about 2 1/2 hours.
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9
To use the duck at once, remove it from the fat and set it and the fat aside to cool to room temperature.
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10
Pick the duck meat from the bones, trying to keep the meat in large pieces.
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11
Discard the bones and skin.
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12
Use now or cover and refrigerate for up to 2 days.
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13
Transfer the fat to an airtight container and refrigerate until solid.
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14
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.
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15
(For longer storage, leave the duck on the bone and submerged in the fat.
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16
Cool to room temperature and then cover and refrigerate for up to 1 week.
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17
When ready to use, reheat in a 250 degree F oven until the fat is liquid and the duck is just warm.)
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18
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.
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19
Set aside until cool enough.
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20
While the potatoes are still warm, fold in the eggs and salt.
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21
Stir in 3/4 cup flour and then add the remaining flour a little at a time until the mixture resembles soft biscuit dough.
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22
The dough should be a little sticky, although when poked with a fingertip the dimple should not close up.
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23
To test the dough, bring a small saucepan of salted water to a simmer.
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24
Break off a 3/4 inch square piece of dough and drop it into the water.
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25
If the dumpling breaks apart in the water, add a little more flour to the dough.
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26
Pour the dough onto a lightly floured work surface and cut into 4 equal pieces.
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27
Line a rimmed baking sheet with parchment paper and lightly dust it with flour.
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28
Working with one piece at a time, roll the dough into a 3/4-inch-thick rope.
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29
Cut each rope crosswise into bite-size lengths, 1 to 1 1/2 inches.
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30
Roll each piece on a gnocchi board or the back of the tines of a fork to ridge them lightly, if you wish.
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31
(The ridges help the dumplings hold the delicious broth.)
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32
Spread the dumplings onto the prepared baking sheet.
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33
Bring a large pot of water to a simmer.
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34
Salt generously.
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35
Set a wire rack inside a rimmed baking sheet and place them next to the stove.
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36
Add half of the dumplings to the boiling water and cook them until they float.
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37
Use a wire skimmer to transfer them to the wire rack.
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38
Cook the remaining dumplings.
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39
The dumplings can be used now as part of the finished dish, or they can be stored to use later.
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40
To store them, refrigerate the dumplings on the wire rack until chilled.
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41
Toss the cold dumplings with enough oil to keep them moist (to prevent sticking).
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42
Transfer to an airtight container and refrigerate for up to 2 days.
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43
To poach the eggs, fill a large skillet with water to a depth of 2 inches.
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44
Add the vinegar and bring to a simmer.
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45
Prepare a large bowl of very cold water.
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46
Break 1 egg into a small bowl or cup and slide the egg into the simmering water.
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47
Repeat with each remaining egg, spacing them evenly in the skillet; cook them in 2 batches if necessary.
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48
Poach at a bare simmer until the whites are firm but the yolks are still runny, about 2 minutes.
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49
Use a slotted spoon to transfer the eggs to the cold water, taking care to not break the yolks.
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50
Use at once or refrigerate until ready to assemble the dish, up to 1 day.
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51
Set the skillet of water aside.
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52
To assemble the dish, in a very large skillet, heat 1 tablespoon of the duck fat over medium-high heat until shimmering hot.
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53
Add half the dumplings and cook until they are golden brown on the bottom, about 3 minutes.
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54
Transfer to a plate.
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55
Heat the remaining 1 tablespoon of duck fat and cook the other half of the dumplings.
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56
Return all of the dumplings to the skillet.
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57
Add the duck and the chicken stock and cook over medium-high heat until the duck is warmed through, 3 to 5 minutes.
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58
Season with the pepper and taste for salt.
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59
Meanwhile, bring the water used to poach the eggs back to a bare simmer.
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60
Divide the duck, dumplings, and broth among 6 warmed shallow bowls.
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61
Working with 1 egg at a time, slip it back into the simmering water for 30 seconds.
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62
Lift each egg from the water with a slotted spoon and pat the bottom dry with a clean kitchen towel.
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63
Place 1 egg on top of each serving.
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64
Sprinkle with thyme leaves and serve at once.
