Stir together the milk, melted butter, egg, and yeast in the bowl of a stand mixer. Add the flour, sugar, and salt and stir until a shaggy dough comes together.
Knead on high for 3-5 minutes or until the dough is smooth and incorporated. Alternatively, mix and knead by hand for about 10 minutes.
Gather the dough into a ball and place in a lightly oiled bowl and cover. Let rise for 2 hours, or until doubled.
Tip the dough out onto a lightly floured work surface and roll to about 1 inch thick. Cut into doughnuts, gathering the excess dough and cutting into doughnuts (or donut holes).
Place the doughnuts on a parchment paper lined baking sheet and brush with melted butter. Let rise for 45 minutes, or until doubled and puffy.
While the doughnuts are rising, make the glaze by mixing together the melted butter, milk, vanilla, and powdered sugar until smooth.
Preheat your air fryer to 350°F. Gently oil the basket and place the doughnuts, slightly spaced out, into the basket. Air fry for 3 minutes. They should will be golden brown and puffy.