Stir together water, yeast, and 1/2 teaspoon of the granulated sugar in a small bowl; let stand until foamy, about 5 minutes. Combine flour, salt, and remaining 1/4 cup granulated sugar in a medium bowl. Add yeast mixture, milk, butter, and egg; stir with a wooden spoon until a soft dough comes together. Turn dough out onto a lightly floured surface and knead until smooth, 1 to 2 minutes. Transfer dough to a lightly greased bowl. Cover and let rise in a warm place until doubled in volume, about 1 hour. Turn dough out onto a lightly floured surface. Gently roll to 1/4-inch thickness. Cut out 8 doughnuts using a 3-inch round cutter and a 1-inch round cutter to remove center. Place doughnuts and doughnuts holes on a lightly floured surface. Cover loosely with plastic wrap and let stand until doubled in volume, about 30 minutes. Place 2 doughnuts and 2 doughnuts holes in single layer in air fryer basket, and cook at 350°F until golden brown, 4 to 5 minutes. Repeat with remaining doughnuts and holes. Whisk together powdered sugar and tap water in a medium bowl until smooth. Dip doughnuts and doughnut holes in glaze; place on a wire rack set over a rimmed baking sheet to allow excess glaze to drip off. Let stand until glaze hardens, about 10 minutes.