1.  In a small saucepan over medium heat, heat milk until warm to the touch.  Transfer to the bowl of a mixer fitted with a dough hook, and add yeast.  Let yeast soften 1 to 2 minutes, then stir to combine. Add 2 3/4 cups flour, without stirring.  Cover bowl tightly with plastic wrap, and set aside in a warm place until the surface of the flour cracks, 30 to 40 minutes.

2.  In a large mixing bowl, whisk together egg yolks, sugar and salt.  Whisk in 1/4 cup flour, and set aside.

3.  In a small saucepan, over medium heat, melt butter.  Split vanilla bean lengthwise, and scrape out pulp and seeds into butter.  Swirl pan as butter heats, and cook 3 to 5 minutes, until bubbles subside and liquid is dark brown.  Remove vanilla bean pods.

4.  Whisk butter into egg mixture.  Stir in vanilla.  Add to yeast mixture.  return to mixer and mix on low for 1 minute.  Add remaining 2 tablespoons flour, and mix to combine. Turn mixer up to medium-high for 1 more minute.  Dough should be very sticky.

5.  Sift a light coating of flour onto a work surface.  Scrape dough onto it, and gather into a ball.  Clean mixing bowl, and lightly coat with oil.  Add dough, cover and allow to rise 1 1/2 to 2 hours, until doubled.

6.  Turn dough out onto lightly floured surface, tuck edges to deflate and return to oiled bowl. Cover and allow dough to rise until doubled, about 30 minutes.

7.  In a deep, heavy pan, heat shortening to 365 degrees.  Turn dough onto lightly floured surface, and gently pat into a rectangle about 1/2-inch thick, flouring surface of dough as necessary.  Dip a 2-1/2 inch doughnut cutter in flour, and cutting as close as possible, cut out doughnuts.  Let doughnuts and holes rest 15 minutes.  Gather scraps of dough together, gently roll or pat to 1/2-inch thickness and cut out remaining doughnuts.

8.  Add doughnuts to pan, a few at a time.  When they rise to the surface, turn them over.  Adjust heat to keep oil at 365 degrees F.  Fry until browned, about 30 to 50 seconds, then flip once more and brown other side.  Drain on paper towels.  Fry remaining doughnuts.  Fry holes separately;  they will take less time.  Cool, then sift powdered sugar over doughnuts.  Or, dip them in coatings.

COATINGS:

Honey Glaze:  In a pan, heat 3/4 cup of honey until it is warm to the touch, and then dip the warm doughnuts in to coat.

Cinnamon Sugar: Combine 1/2 cup sugar with 1 teaspoon salt and 1 1/4 teaspoons cinnamon, and dip the warm doughnuts in to coat.

White Glaze:  Combine 1/2 cup plus 2 tablespoons sifted powdered sugar in the top of a double boiler over simmering water with 1/4 cup heavy cream, 1/2 teaspoon vanilla extract and 1/8 teaspoon salt.  Heat until just warm, stirring frequently, until thin and translucent.  Use a pastry brush to cover doughnuts.

Recipe from Nancy Silverton's Pastries from La Brea Bakery", Villard