Melt the butter for 1 minute in the microwave. It should be all the way melted. Let it cool for 5 minutes. Then, heat the milk in a microwave safe bowl for 1 minute. The milk should be warm at 105F, but not hot.
Using a mixer with a paddle attachment, add the milk, yeast, slightly beaten egg, melted butter, vanilla, salt, and sugar and mix until combined. Switch to a dough hook, then add 3 cups of flour. Mix on low until flour starts to incorporate, then switch to medium and mix until combined. Let it sit for 5 minutes.
Turn the mixer to medium and let it mix for 5 minutes. If your dough is very sticky, add an extra ⅓ cup at this point. Then mix for another minute until the flour is incorporated. Take it off the mixer and place in a medium bowl that has been sprayed with non-stick spray. Place the dough in the bowl and cover with a kitchen towel. Let it rest for 2 hours until the dough is doubled in size.
Punch the dough with your fist. Then place your dough on a lightly floured surface. Roll out your dough until is ½ inch thick. Then take a doughnut cutter, and cut out 6 doughnuts. Cover the doughnuts and doughnut holes with a kitchen towel for 1 hour.
Start heating the oil in a heavy bottom saucepan on medium heat. Use a kitchen thermometer, and heat the oil until 350F. It takes about 10-15 minutes to heat up. If it gets too hot, turn down the heat by a little bit. Fry the doughnuts 2 at a time. They should be lightly browned. They will need 45-60 seconds per side. Set doughnuts on a plate covered in paper towels.
Once the doughnuts have cooled slightly, transfer to a cooling rack. Allow the doughnuts to cool completely before applying the glaze.