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Preparation Whisk baking powder, 2 cups flour, and 1 tsp. salt in a medium bowl to combine. Whisk egg yolks, yogurt, granulated sugar, butter, and vanilla in a large bowl until smooth. Add dry ingredients to egg mixture with a spoon or rubber spatula until dough comes together and is smooth with no spots of dry flour remaining.
Turn out dough onto a well-floured large sheet of parchment or wax paper (you can be generous with the flour since you’ll brush off the excess later). Dust dough with more flour, place another large sheet of parchment or waxed paper over, and roll to ½" thick. Remove top sheet of parchment and brush off any excess flour with a dry pastry brush.
Punch out circles with a 3½"-diameter biscuit or cookie cutter, glass, or mug. Punch out centers of rounds with a 1"-diamter cutter or plastic twist-off bottle cap (or poke a hole with a chopstick, then gently pull hole open to widen the center). Reroll any scraps and punch out additional doughnuts and holes.
Fit a large pot with thermometer; pour in oil to come 1½" up the sides of the pot. Heat over medium-high until thermometer registers 350°F. Fry doughnuts, turning halfway through, until puffy and a color somewhere between golden brown and mahogany, about 4 minutes total (time will be slightly less for doughnut holes). Using a spider or slotted spoon, transfer to a wire rack and let cool 10 minutes.
Meanwhile, whisk lemon zest, powdered sugar, a tiny pinch of salt, and ¼ cup water in a small bowl until a smooth glaze forms.
Dip each side of doughnuts and holes into glaze, letting excess run off before returning to rack. Eat as soon as you can’t stand the suspense anymore.