Whisk together 2 cups all-purpose flour, 2 teaspoons baking powder, 1/4 teaspoon salt, and 1/4 teaspoon ground nutmeg in a large bowl until evenly combined.

In another bowl, whisk 3/4 cup whole milk, 1 large egg, 1 teaspoon vanilla extract, and 2 tablespoons vegetable oil until smooth.

Pour the wet ingredients into the dry ingredients and stir gently with a whisk until just combined, taking care not to overmix.

Cover the bowl with plastic wrap and refrigerate the dough for 15 minutes to chill and firm up.

Heat 2 quarts of vegetable oil in a deep skillet or Dutch oven to 350°F, monitoring with a candy thermometer.

Lightly flour a work surface and roll or pat the chilled dough to 1/2-inch thickness, then cut out doughnut rings and holes with a doughnut cutter.

Carefully fry 3 to 4 doughnut rings at a time in the hot oil for 1 to 1½ minutes per side, until golden brown; fry holes for about 45 seconds per side.

Use a slotted spoon to transfer fried doughnuts and holes to a paper towel-lined plate to drain excess oil and cool for 2 to 3 minutes.

While doughnuts cool slightly, whisk together 2 cups powdered sugar, 3 tablespoons milk, 1/2 teaspoon vanilla extract, and 2 tablespoons melted butter until smooth.

Dip the tops of slightly warm doughnuts into the glaze, twisting to coat evenly, then place on a wire rack over a baking sheet to let excess drip off.