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1
TO MAKE THE DOUGH, combine the yeast and 1/2 cup of the milk in a bowl, stirring until the yeast is dissolved.
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2
Let sit until it begins to foam, about 5 minutes.
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3
Combine 4 cups of the flour with the sugar in a mixer with the hook attachment and add the foamy yeast mixture.
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4
Scrape the vanilla bean into the mixture and add the remaining 3/4 cup milk, the salt, eggs, and egg yolks.
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5
Discard the vanilla bean or save for another use.
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6
Mix on low speed until the dough starts to come together, and then add the butter.
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7
Increase the speed to medium and mix until the dough is smooth and elastic, 15 to 20 minutes.
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8
It will be slightly sticky but shouldnt tear easily.
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9
Lightly coat a large bowl with canola or vegetable oil.
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10
Put the dough inside and cover with a slightly damp towel or plastic wrap.
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11
Let rise in a warm area (about 70F) until doubled in size, about 1 1/2 hours.
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12
Once its doubled, bring the edges to the center and turn over.
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13
Turn the dough out onto a lightly floured surface.
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14
Roll out into a rectangle about 1/2 inch thick and cut out as many 2 1/2-inch circles as possible.
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15
Put them on a parchment paperlined baking sheet and cover lightly with a towel.
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16
Gather the scraps and put them on top of each other instead of making a big ball.
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17
Reroll the dough (if you feel like the dough is shrinking as youre trying to roll it out, wrap it in plastic and refrigerate for about 10 minutes) and cut out more circles for a total of about 42.
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18
Leave a few pieces of dough of the same thickness to test the oil later.
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19
Cover with a towel and let rise in a warm area (about 70F) until doubled in size and the dough springs back when touched, about 30 minutes.
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20
TO MAKE THE STRAWBERRY JAM, slice or coarsely chop the strawberries and put in a small nonreactive pot with the sugar and lemon juice.
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21
Cook over medium heat, stirring, until the sugar dissolves.
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22
Add the salt and adjust the heat to maintain a constant soft boil and skim off the foam as it forms.
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23
Continue cooking until the jam tightly grabs a spoon or sets when you put a little bit on a cold plate, 15 to 25 minutes.
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24
Pour the oil into a heavy pot to a depth of 3 inches and heat to 375F (use a piece of the dough you set aside to test the oil; it should sink and then quickly rise to the top and bubble).
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25
Fry the doughnuts, a few at a time, turning with a slotted spoon, chopsticks, or tongs, until evenly golden on both sides and puffy, 2 to 3 minutes.
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26
Drain on a paper bag or towel and roll in sugar while still warm.
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27
Fit a pastry bag with a small flat tip (about 1/2 inch) and spoon in the strawberry jam.
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28
Insert the tip into the side or bottom of each doughnut and fill with jam.
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29
Serve immediately.
