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1
Put the potatoes in a medium pot and cover with cold water; season with salt.
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2
Bring to a simmer over medium-high heat and cook until fork-tender, about 20 minutes; drain.
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3
Pass the potatoes through a ricer into a large bowl or mash with a potato masher until smooth.
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4
Combine the milk, butter, 1/2 teaspoon salt and 1/4 cup water in a medium saucepan and bring to a boil.
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5
Remove from the heat, add the flour and stir vigorously with a wooden spoon until incorporated.
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6
Return the saucepan to the stove and cook over low heat, stirring, until the mixture comes together and pulls away from the side of the pan, about 1 minute.
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7
Continue stirring until the mixture is slightly shiny, about 4 more minutes.
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8
Transfer the mixture to a stand mixer and beat with the paddle attachment on medium-high speed, about 1 minute.
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9
Add half of the mashed potatoes and the whole egg and beat until incorporated.
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10
Add the remaining potatoes and the egg yolks and beat until creamy and smooth, about 1 more minute.
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11
Transfer the mixture to a pastry bag fitted with a 1/2-inch round tip.
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12
Set a large piece of parchment paper on a cutting board.
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13
Pipe about 24 mounds (1 to 1 1/2 inches each) of the potato mixture on the parchment.
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14
Cut the parchment between the mounds with a paring knife so each puff sits on its own piece of paper.
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15
Heat 2 inches of peanut oil in a large pot until a deep-fry thermometer registers 325 degrees F. Working in batches, carefully invert each puff into the oil, leaving it on the parchment.
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16
The parchment will fall off as the puffs cook; remove with tongs.
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17
Fry until golden, turning as needed, 6 to 8 minutes.
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18
Remove with a slotted spoon and drain on paper towels; season with salt.
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19
Photograph by David Malosh