Line a plate with paper towels, and place a wire rack inside a large rimmed baking sheet; set aside. Pour oil to a depth of 2 inches in a large Dutch oven; heat over medium-high until a deep-fry thermometer registers 360°F, about 20 minutes.

Meanwhile, sift flour, baking powder, salt, nutmeg, ½ cup of the sugar, and 1 teaspoon of the cinnamon over a large bowl; stir gently to combine. Stir in melted butter until crumbly. Stir in milk and beaten egg until smooth.

Turn out dough onto a lightly floured surface. Knead dough until smooth and elastic, about 1 minute.

Roll dough to ¼-inch thickness. Using a 3 ½-inch donut cutter (or a 3 ½-inch and a 1 ¼-inch round circle cutter), cut out 12 donuts, rerolling scraps and donut holes as needed.

Whisk together remaining 1 cup sugar and 1 teaspoon cinnamon in a shallow bowl.

Working in 4 batches, add donuts in an even layer to hot oil, adjusting heat as needed to maintain 350°F. Cook, turning once halfway through, until golden brown, about 2 minutes total per batch. Let donuts drain on prepared plate for 30 seconds.

Working with 1 warm donut at a time, place donut in cinnamon-sugar mixture, turning to fully coat; place coated donuts in an even layer on prepared wire rack. Repeat process with remaining warm donuts. Repeat frying and coating procedure with remaining donuts. Serve donuts warm or at room temperature.