Chocolate Frosted Doughnuts

1 cup all-purpose flour

1/3 cup granulated sugar

1 teaspoon baking powder

1/4 teaspoon baking soda

1/2 teaspoon salt

2 teaspoons vanilla extract

1 large egg, brought to room temperature

1/2 cup whole milk

2 tablespoons unsalted butter, melted and cooled

Chocolate Glaze

1 1/2 cup confectioner’s sugar, sifted

4 tablespoons cocoa powder, sifted

4–5 tablespoons whole milk

2 teaspoons vanilla extract