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1
Heat the whole milk in a small saucepan over medium-low heat, to 110u00b0 F to 115u00b0 F.
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2
Dissolve the yeast in the warm milk in a medium bowl. Add 1 tablespoon of the sugar and 1/2 cup of the flour, whisk together thoroughly, and set aside.
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3
In a small saucepan or in a small, microwave-safe bowl in a microwave oven, melt the butter and set aside to cool.
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4
In a separate small bowl, vigorously whisk together the remaining 4 tablespoons and 1/2 teaspoon sugar, egg yolks, and rum until frothy.
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5
Place the remaining 5 3/4 cups flour in a large bowl. Whisk the yeast mixture again, and pour it over the flour. Add the egg mixture, and mix with a wooden spoon or a spatula until the dough just starts to come together. Add the melted butter, and combine until smooth.
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6
Liberally flour a work surface, and turn the dough out onto it. Knead the dough until it comes together and no longer sticks to your hands when worked. If the dough seems dry, add a little more milk and knead. If the dough seems too wet, add a bit more flour.
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7
Lightly oil a large bowl and place the dough in it. Cover with a towel or plastic wrap, and let rise in a warm area until it doubles in size (about 1 hour).
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8
Punch the dough down and separate it into 2 balls. Flour your work surface again, and roll the first ball into a disk about 1/2-inch thick.
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9
Using a 3-inch cookie cutter or an inverted drinking glass, cut rounds out of the dough. Set the scraps of dough aside.
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10
Place 1 tablespoon prune butter in the center of 1 of the rounds. Top it with another round, and pinch the seams with your fingers to seal the edges. Then bring together the edges of the dough on 1 side to create a sphere (see process below), and pinch to make a new seam. Reshape each ball with your hands to re-form it into a fluffy round shape. Repeat this process for the remaining rounds. Set each filled paczki on a floured surface.
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11
Repeat this process with the remaining ball of dough. Reroll all the scraps of dough, and repeat this process again.
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12
Let the filled paczki rise for 20 minutes, or until fluffy.
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13
After the paczki have risen for 15 minutes, heat the oil in a heavy-bottomed steel or cast-iron skillet to 350u00b0 F on a candy thermometer, and line a large plate or a cooling rack with paper towels.
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14
Once the oil is ready, use a slotted spoon to carefully place 3 or 4 paczki in the hot oil. Fry for 45 seconds, or until golden brown on 1 side. Flip the paczki and fry on the other side until golden brown, about 25 to 35 seconds.
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15
Remove the paczki from the oil immediately and transfer to the prepared plate or cooling rack.
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16
Let the paczki cool completely. Using a fine-mesh sieve, sprinkle them with confectioners' sugar. Serve immediately.
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17
Note: To ensure your paczki are perfectly fried but not greasy, heat your oil to 350u00b0 F before frying and reheat it to 350u00b0 F between batches.
