In a microwave-safe bowl, add in the milk and butter. Microwave for 45 seconds and stir. Check the temperature of the heated milk and make sure it's about 110 degrees. If it's too cold, microwave for another 10 - 15 seconds.
Add in the package of yeast.
Measure out the 1/2 cup of granulated sugar. Sprinkle about 2 teaspoons of sugar in with the yeast. Give it a stir and let it sit for about 5 - 10 minutes. The yeast mixture will foam.
In a large glass bowl, add in the remaining sugar and 2 cups of all purpose flour. whisk to incorporate the sugar and flour. Make a well in the center of the flour.
Add the yeast mixture, egg, and vanilla into the well. Using a whisk, mix all the liquid ingredients until combined. Try not to get into the flour yet.
Switch to a spatula, and stir the flour into the wet mixture until incorporated. It will still be very sticky.
Sprinkle 2 tablespoons of flour on your clean work surface (I sprinkled mine on parchment paper). Place the sticky dough on in the center of the flour. Carefully knead the flour you sprinkled down into the dough. I only kneaded mine about 10 times.
Spray the glass dish you used to mix the dough together in with non-stick cooking spray or 1 tablespoon of vegetable oil.
Place the dough in the bowl and spray the top of the dough with the non-stick cooking spray. If you used vegetable oil, roll the dough in the oil so it's covered.
Place the bowl in a warm place and cover with a tea towel until it doubles in size (about 2 hours).
Once the dough has doubled, sprinkle the 2 tablespoons of flour down on your clean work surface.
Place the dough on the surface and sprinkle the tops with some of the flour. Press the dough down until it's about 1/2 inch thick.
Measure out two 10" by 10" pieces of tin foil. You'll place 3 cut-out donuts on each piece.
Using a 3 1/2 inch donut cutter or a glass around the same size, cut out the donuts and place on the tinfoil. Space out the donuts so as they rise they won't touch each other.
Cover the donuts with loosely fitting plastic wrap and place them in a warm place. Let them rise for 30 minutes.
Place one of the pieces of tinfoil with the donuts on it on the lowest convection oven rack. Place it in the oven.
Set the convection oven to 300 degrees. Let the donuts bake for 10 minutes. The donuts will start to get brown.
Rotate the donuts so the pail sides are inside and browning. Let it bake for another 3-5 minutes. Be sure to watch them and make sure they aren't getting too brown.
Repeat with the other donuts.
Cinnamon Sugar Coating
In a plate or very shallow bowl, add in the cinnamon and sugar. Stir to combined.
In a microwave-safe dish, microwave the butter for 30 seconds.
Brush the butter on the tops of the hot donuts. Dip the tops of the donuts into the cinnamon and sugar mixture.
Repeat with all the donuts.
Let the donuts cool 15 - 20 minutes.
In a shallow bowl, add in the powdered sugar, vanilla, and milk. Whisk until smooth.
Dip the cooled donuts into the glaze and hold it upside down so the extra glaze will drip off.
Repeat if you want an extra layer of glaze.
Immediately add the sprinkles and let dry for about 30 minutes.