Measure and add the 1/2 teaspoon sugar and 1 teaspoon yeast into a medium bowl. Stir in warm (100-110°) water. Let sit for 5 to 10 minutes.
Meanwhile, in a large bowl, add the flour, remaining 1/4 cup sugar and kosher salt. Stir to combine.
Next add in the yeast mixture, milk, unsalted butter and egg. Using a wooden spoon, stir until a soft dough forms.
On a lightly floured surface, turn out the dough and kneac until the dough becomes smooth. About 1 to 2 mintues.
In a lightly oiled bowl, add the dough and cover lightly with a tea towel. Place the bowl in a warm spot to rise for 1 hour.
Once risen, turn the dough out onto a floured surface and roll to 1/4 of an inch thick. Use a 3-inc round cutter and stamp out 8 doughnutes. Use a 1-inch round cutter to stamp out the middle of each doughnut.
Place the doughnuts and holes on a lightly floured surface, losely cover with plastic wrap and let rise a second time for 30 minutes.
Spray the basket of your air fryer with olive oil and preheat your air fryer to 350-360°. Working in batches air-fry 2 doughnuts and 2 doughnut holes for 4 to 5 minutes or until golden. Transfer to a wire rack and repeat with remaining doughnuts and holes.
Lastly, in a medium bowl, whisk together the powdered sugar and 4 teaspoons water until smooth.
Use forks to dip the doughnuts in the glazed. Place on a wire rack set into a pan to catch any glaze that drips off. Repeat and eat.