Add the milk, yeast, and sugar to the bowl of a stand mixer. Stir and then let it bloom for a few minutes.
1 cup milk, 2¼ teaspoons active dry yeast, 3 tablespoons granulated sugar
Whisk in the eggs, vanilla, and salt. Attach the dough hook to the mixer. Mix on low speed while gradually adding in the flour.
2 large eggs, 1 teaspoon pure vanilla extract, 1½ teaspoons kosher salt, 3¾ cups all-purpose flour
Continue to mix on low for 3-4 minutes until the dough becomes smooth and elastic and pulls away from the sides of the bowl. If the dough doesn’t pull away from the bowl, add in a couple more tablespoons of flour at a time and continue mixing.
Remove from the dough from the mixer and place on the counter to knead a few times and shape into a ball.
Wrap in plastic wrap and chill in the refrigerator for 30 minutes.
On a floured surface, roll out the dough to a 12x18-inch rectangle. Visually divide the dough into thirds.
Spread 7 tablespoons of butter onto the middle portion of the dough.
¾ cup unsalted butter
Fold one side on top of the buttered middle third. Spread the rest of the butter on top of the folded side.
Fold the remaining side inward on top of the 2nd buttered portion.
Place the dough on a baking sheet, cover with plastic wrap, and refrigerate for another 30 minutes.
Roll out the dough into a rectangle about the same dimensions as before. Fold the dough using the same pattern as before, turning each third inward over the middle third.
Place on the sheet pan, cover and refrigerate for 30 minutes.
Repeat the process of rolling to a rectangle and folding inward 2 more times.
Place the dough back on the sheet pan, cover, and refrigerate for at least 1 hour.
Roll out the dough to a little under ½-inch thick.
Use a donut cutter or a round cookie cutter or jar to cut out 8-10 circles. Use a smaller round cookie cutter to cut out the holes in the circles.
Place the donuts and donut holes on a lined baking sheet.