Combine the warm milk and yeast in the bowl of a stand mixer fitted with the dough hook attachment. Allow the yeast to sit for five minutes, or until it has become foamy on top.
Next, add the egg, melted butter, and coffee extract to the bowl and mix until combined.
With the mixer on low, slowly add the flour followed by the brown sugar.
Turn the mixer to medium speed and knead the dough until it easily pulls away from the sides of the bowl. It should be smooth, but pretty sticky.
Transfer the dough to a lightly oiled bowl and place it in a warm spot to rise until doubled in size, about an hour.
Once the dough has risen, transfer it to a lightly floured surface and roll out to 1/2-inch thick. Use round cutters to cut out the doughnuts. (see note)
Transfer the donuts to the air fryer basket, making sure not to overcrowd them. Cook at 400F for 7 minutes, or until the donuts are golden brown on the outside.
Whisk the icing together and set aside while the donut cook. Set aside.
Remove the donuts from the air fryer and allow them to cool slightly on a wire rack before icing.
Once cooled, spread the icing over the donuts and top with sprinkles, if desired. Store the donuts in an airtight container at room temperature for up to two days.
If you don't want to buy a donut cutter, you can use a 2.5-inch circle cookie or biscuit cutter to cut out the donuts. Then use the wide end of a piping tip to cut out the holes. (You can also air fry the donuts holes!)