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1
In the bowl of a stand mixer fitted with the paddle attachment, stir together 2 cups of the flour, the yeast, sugar, and salt.
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2
Add the warm milk, vanilla, and egg yolks.
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3
Mix until smooth.
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4
Add the remaining flour and the butter and mix until incorporated.
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5
Continue to mix on medium speed until the dough is soft and smooth (it will be slightly sticky).
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6
Cover the bowl with plastic wrap and set in a draft-free place to rise for 1 hour, or refrigerate for up to 12 hours.
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7
Lightly flour 2 baking sheets.
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8
Roll out the dough on a lightly floured surface and cut with a floured donut cutter or a 3- to 3-1/2-inch biscuit cutter (for the latter, youll also need a 1-1/2-inch biscuit cutter to form the donut holes).
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9
Discard the dough scraps, or cut into pieces to fry separately after the donuts are done, as snacks.
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10
Place the donuts and donut holes at least 1 inch apart on the baking sheets and cover loosely with plastic wrap.
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11
Set in a warm place to rise a second time until they are puffy and nearly double in size, about 15 to 30 minutes.
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12
The donuts are ready when the dough springs back slowly after being touched with a fingertip (if it springs back at once, it needs more time; if it collapses, it has overproofed and you can reroll the dough and cut it once).
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13
While the donuts rise, fill a large, heavy-bottomed pot with at least 2 inches of oil (the oil should not come more than halfway up the sides of the pan).
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14
Heat over medium-high heat until a deep-frying thermometer registers 360 degrees F. Using a spatula, carefully lower a few donuts into the oil; do not crowd the pot.
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15
Fry for 1 to 2 minutes per side, turning a few times, until the donuts are a light golden brown and cooked through.
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16
Remove with a spider or slotted spoon, drain on a wire rack over a paper towel, and let cool slightly before glazing or tossing in sugar.
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17
Repeat with the remaining donuts and donut holes.
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18
Be sure to keep the temperature consistent while frying.
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19
Serve the donuts as is (warm is best), tossed in powdered sugar when cool, or glazed.
