Matcha Doughnuts

Whisk granulated sugar, baking powder, salt, and 2 cups flour in a large bowl. Make a well in the center and add egg, buttermilk, and butter. Using a wooden spoon or a spatula, gradually incorporate dry ingredients until a shaggy dough forms. Turn out onto a lightly floured surface and knead a couple of times or so until no wet spots remain (be careful not to overwork or doughnuts will be tough).
Roll out dough to 1/4" thick, sprinkling surface and/or dough with flour as needed to prevent sticking. Punch out as many rounds as you can with 3 1/4" cutter, then use 1 1/4" cutter to punch out center of each round. Gather doughnut holes and scraps and re-roll; repeat process until all dough has been used (3-4 times; you should have 8 doughnuts).
Fit a medium saucepan with thermometer. Pour in oil to come 4" up sides of saucepan and heat over medium until thermometer registers 325u0b0F. Working in batches, fry doughnuts until deep golden brown, about 3 minutes per side. Transfer to a wire rack lined with paper towels and let cool slightly.
Whisk together 1 cup granulated sugar and 1 tablespoon matcha until no lumps remain.
Whisk together 1 cup powdered sugar and 1 tablespoon matcha until no lumps remain.
Whisk together 1 large egg white, 2 cups powdered sugar, 2 teaspoons matcha, a pinch of salt, and 1 tablespoon water until smooth.
Toss still-warm doughnuts in Matcha Sugar Topping or Sweet Matcha Powder or dip in Matcha Glaze.