Stir together flour, baking powder, and salt in a bowl. In a large bowl, beat eggs, sugar, and vanilla with an electric mixer at medium speed until thick, about 3 minutes. In a small bowl, stir together milk and melted butter.

Add flour mixture and milk mixture alternately to the egg mixture, beating at low speed after each addition just until combined. Chill dough, covered, until easy to handle, about 1 hour.

Roll dough until 1/2-inch thick on a lightly floured surface. Cut donut holes using a 1½-inch round cutter. Dip cutter into flour between cuts and reroll dough as needed. Set a wire rack inside a sheet pan.

Pour oil to a depth of 2½ to 3 inches in a 4-qt. pot or deep-fat fryer. Heat to 365°F using a thermometer to monitor temperature.

Working in batches of 5 to 6 donut holes, fry, turning once halfway through, until light and golden, about 2 minutes.

Transfer donut holes to the prepared rack and let cool about 2 minutes.

Whisk powdered sugar, melted butter, vanilla extract, and salt in a bowl to make the vanilla glaze. Whisk in hot water, 1 tablespoon at a time, until the desired consistency.