Using a stand mixer with a paddle attachment or a hand mixer, beat eggs and sugar until light and fluffy.
Add vanilla and the melted butter and beat until combined.
In a separate bowl, whisk together flour and baking powder. Add 1/2 of the dry ingredients and 1/2 of the milk to the bowl and beat. Add in the other 1/2 of the dry ingredients and the rest of the milk and beat until you get a smooth, silky batter. If the batter seems really thick, then add 1 or 2 tbsp of milk to thin out the batter slightly.
Spoon batter into donut pans. For donuts with holes, I use a piping bag and pipe in batter in a circular motion about 2/3 full. For donuts with no holes, fill the pan 2/3 of the way full (don't worry if the hole is covered, this will give you a more more surface area to frost! Kid pleaser!)
Bake 12-15 minutes or until donuts start to brown around the edges and a toothpick comes out clean.
Remove donuts and let cool in the pan for about 5 minutes.