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Whisk together flour, granulated sugar, baking powder, salt, baking soda, and nutmeg in a large bowl until combined. Whisk together egg, egg yolk, buttermilk, butter, and vanilla in a medium bowl until combined. Fold buttermilk mixture into flour mixture, folding just until a soft sticky dough forms (do not over stir). Cover bowl with plastic wrap, and refrigerate until chilled, about 1 hour.
Add oil to a large Dutch oven to a depth of 4 inches, and heat oil over medium until a thermometer registers 350°F.
Meanwhile, roll chilled dough to 1/2-inch thickness on a lightly floured work surface. Using a lightly floured 3 1/2-inch round cutter, punch out as many rounds as you can. Using a lightly floured 1 1/4-inch round cutter, punch holes out of each round to make a donut shape. Reserve smaller dough rounds to make donut holes. Gather dough scraps, and gently reroll 1 time, repeating cutting procedure. You’ll have 10 donuts and 10 donut holes total.
Set a wire rack inside a rimmed baking sheet. Line a second baking sheet with paper towels. Working with 2 to 3 donuts or donut holes at a time, add donuts to hot oil; cook until golden brown and cooked through, about 2 minutes per side for large donuts and 1 minute per side for donut holes. Using a spider, transfer cooked donuts to prepared wire rack, and let drain about 1 minute. Transfer drained donuts to paper towel-lined baking sheet, and let cool about 10 minutes. Repeat frying and draining procedure with remaining donuts and donut holes.
Whisk together powdered sugar, lemon juice, and water in a medium bowl until smooth. Dip 1 side of each donut into glaze, letting excess glaze fall back into bowl. Place glazed donuts on wire rack; let stand until set, about 10 minutes. Serve. Donuts may be stored, covered, at room temperature up to 1 day.