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 325°F. 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.