Add the milk, yeast, and sugar to the bowl of a stand mixer fitted with the dough hook attachment (or into a large bowl in mixing by hand). Mix until well combined.
Next, add the egg, vegetable oil, and vanilla bean paste and mix well.
Add the flour gradually with the mixer on low speed. Continue kneading on low speed until the dough is smooth and slightly tacky-- about 5 minutes.
Place the dough in a greased bowl, cover in plastic wrap, and set it aside to rise for one hour.
Roll the dough out on a lightly floured surface until it is ½” thick. Use a doughnut cutter (or a biscuit cutter and a small circular object such as a piping tip) to cut out 3-inch doughnuts. Set the doughnuts (and doughnut holes, because why not?) on a silicone or parchment-lined sheet pan to rise for an additional hour.
Grease basket of air fryer with non-stick spray. Air fry in batches, a few at a time, at 365 F for 7-8 minutes, or until they are a deep golden brown. Place on a cooling rack and repeat with the remaining dough.
For the Vanilla Bean Icing
Combine the powdered sugar, heavy cream, and vanilla bean paste in a small bowl.
Dip the doughnuts in the icing, and place them on a wire rack to set. Serve warm.