Vanilla-Glazed Doughnuts

Make the brioche dough: Mix the dough as directed.

Shape the doughnuts: Line two sheet pans with 18 3-inch (7.5 cm.) squares of parchment paper. Dust a work surface with some tapioca starch. Knead the dough a few times, then roll to 3/4-inch (2 cm.) thickness.

Dip a 2 1/2-inch (6.5 cm.) cookie cutter in some tapioca starch and cut the dough into rounds. Use a 3/4-inch (2 cm.) round cookie cutter to cut a center circle out of each round of dough. Place the cut doughnuts on the parchment squares. Cut all the dough and knead any remaining scraps together, then roll and cut the scraps the same way.

Proof the dough: Loosely cover both sheet pans with a linen towel or plastic wrap and proof until the dough feels light and marshmallow-like, about 40 to 45 minutes.

Make the vanilla glaze: In a medium bowl, whisk together the powdered sugar, warm water, vanilla, and salt. Set aside.

Fry the doughnuts: About 20 minutes before the dough is done proofing, pour 3 inches (7.5 cm.) of oil into a medium saucepan or dutch oven. Clip a deep-fry thermometer to the side of the pan and heat over medium-high heat to 350°F (180°C). Place a wire rack over a sheet pan next to your pan. Use the parchment ends to lift the dough and carefully invert onto the hot oil. Be careful not to drop the doughnuts in from high up, or the oil will spatter and could burn you. Do not overcrowd the pan. The doughnuts will fall to the bottom and then rise again. Cook for 1 to 2 minutes on each side until golden brown. Use a spider tool or a perforated spatula to scoop the doughnuts out of the oil and place them on the wire rack. Continue cooking the doughnuts.

Glaze the doughnuts: While the doughnuts are warm, dip them in the glaze. Return them to the rack. Do a second dip for a thicker coating. If using any toppings, sprinkle them on while the glaze is wet. Serve the doughnuts warm. They are best eaten the same day.