Lemon juice and zest brighten these rich, dense buttermilk donuts. They're dipped in a lemon glaze and sprinkled with more zest before serving.
Lemon Confectioners'-Sugar Glaze: Whisk together confectioners' sugar and salt. Whisk in juice until mixture is smooth and thick, and has the texture of pourable honey. Glaze can be refrigerated in an airtight container up to 3 days; stir until smooth before using. (Makes 1 1/2 cups.)
Donuts: Dust a baking sheet with flour. Whisk together flour, baking powder, baking soda, and salt. Beat butter, granulated sugar, and 1 tablespoon lemon zest on medium-high speed until light and fluffy, about 2 minutes. Beat in egg and yolk, then lemon juice. Scrape down bowl, then beat in dry ingredients in three additions, alternating with buttermilk, beginning and ending with dry ingredients. Turn out dough onto baking sheet; use floured hands to flatten to a 3/4-inch-thick rectangle. Refrigerate at least 1 hour and up to overnight (if longer than 1 hour, cover with plastic wrap).
Pour oil into a large, deep, heavy pot, such as a Dutch oven, until it reaches 2 inches up side, leaving about 2 1/2 inches headroom. Clip a deep-fry thermometer to pot and heat oil over medium to 350 degrees. Meanwhile, using a floured 2 1/2-inch round donut cutter, cut out donuts (or use a 2 1/2-inch round cookie cutter, then cut out holes with a 1- to 1 1/2-inch round cutter). Pat together scraps; cut out more donuts (these will not be quite as tender). Working in batches of 3 so as not to crowd pot, carefully drop donuts into hot oil and cook, flipping a few times, until deep golden brown all over, 4 1/2 to 5 1/2 minutes (2 1/2 to 3 1/2 minutes for holes), maintaining oil temperature between 340 degrees and 350 degrees at all times. Transfer donuts and holes to a wire rack set in a rimmed baking sheet; let cool 10 minutes.
While still slightly warm, dip top of each donut in glaze until halfway submerged, turning as needed to evenly coat one side, then place, glaze-side up, on rack. Zest lemons directly over donuts. Let stand until glaze sets and donuts are cool, about 1 hour. Donuts are best eaten the day they are made.
If you want to make all donut holes, flatten dough to 3/4 to 1 inch thick. Use a floured 1 1/2-inch cutter to cut out about 42 rounds. Cook, flipping a few times, until deep golden brown all over, 2 1/2 to 3 1/2 minutes total.