Preheat the oven to 350F/175C degrees. Grease 2 doughnut pans well.
Into a large bowl, sift together the flour, sugar, baking powder, and salt.
In a small bowl, whisk together the egg, milk, melted butter, and vanilla.
Stir the wet mixture into the dry ingredients until just combined.
Spoon the batter into the baking pans, filling each one about ½ full.
Bake for 15-17 minutes, until a cake tester comes out clean.
Allow to cool for 5 minutes, then tap the doughnuts out onto a sheet pan and transfer to a rack to completely cool.
Place baking paper under wire racks to catch the excess icing.
For the icing:
Prepare by mixing the sugar, butter and vanilla together in a bowl and then add the milk one tablespoon at a time until the desired consistency is reached (I like mine fairly thick).
Dip the cooled doughnuts into the icing one at a time covering half the doughnuts then place on the cooling racks.
Add the sprinkles before the icing sets.
Leave on cooling rack to completely set the icing before serving.
Notes
Don't overfill the pan. Do a test run with a single donut to make sure they don't rise too much (making an uneven donut).
Use a squeeze bottle to pour the batter this keeps things neat and more precise.
Be sure togrease your pan. Most donut pans are nonstick, but that's not always a guarantee.
Don't have a donut pan? If you'd like one like mine, you can get it here on Amazon. (#paidlink). I highly recommend the two-pack.