1. In a small saucepan set over medium heat, heat the milk until warm to the touch. Place yeast in the bowl of an electric mixer. Pour warm milk over the yeast. Sprinkle 2 3/4 cups flour over the milk, and let stand until the flour begins to crack, about 20 minutes.

2. In a medium bowl, whisk together egg yolks and sugar. Add salt and 1/4 cup flour; whisk to combine; set egg mixture aside.

3. Melt butter in a small saucepan over medium heat. Using a paring knife, scrape vanilla seeds into melted butter; add pod. Cook until butter browns, foams, and has a nutty aroma, about 3 minutes. Remove and discard pod. Pour butter over the egg mixture; whisk to combine. Stir in vanilla extract.

4. Add butter mixture to yeast mixture. Mix on low speed with paddle attachment 1 minute. Add remaining 2 tablespoons flour; combine. Increase speed to medium high; mix 1 minute more. Dough will be very sticky.

5. Lightly oil a large bowl. Sift an even layer of flour onto a clean work surface. Turn out dough, and form a ball. Place in oiled bowl, and cover with plastic wrap. Set bowl aside in a warm place, and let dough rise until doubled in size, about 1 1/2 hours.

6. Turn out dough again, and fold in half, deflating dough. Return dough to oiled bowl, re-cover, and set aside to rise until doubled in size again, about 30 minutes.

7. Heat the oil in a low-sided six-quart saucepan over medium-high heat until a deep-frying thermometer registers 375 degrees. Lightly dust a baking pan with flour, and line a second one with paper towels.

8. Turn out dough; dust with flour. Pat dough into an 8-by-12-inch rectangle, about 1/2 inch thick. Using a 2 3/4-inch doughnut cutter, cut out doughnuts as close together as possible, dipping cutter in flour before each cut. Transfer holes to the floured baking pan.

9. Flour your hands, and make twists out of the rings: Lift one side of a doughnut by placing your thumb on top and your forefinger underneath; reverse placement of other hand on other side. Turn each side away from each other, forming a doughnut with two twists. Transfer twists to floured pan along with holes; set aside 10 minutes, but not longer.

10. Lift a twist at each end, pull to elongate, and carefully place in hot oil. Repeat with three more twists. Cook until golden, about 1 1/2 minutes. Using a slotted spoon, turn over, cook until evenly browned on both sides, about 1 1/2 minutes more. Transfer to the lined pan. Repeat with remaining twists.

11. Drop holes, one at a time, into hot oil; cook until completely golden brown, about 2 minutes, turning with slotted spoon as needed. Transfer to lined pan.

12. Gather remaining dough scraps into a ball. Let rest 10 minutes; pat into a 1/2-inch-thick rectangle. Cut, let rest 10 minutes, and cook.

13. Coat twists and holes with white glaze. Transfer to a wire rack, and let set. Serve.