In the bowl of your stand mixer with the paddle attachment, beat the egg and sugar on medium speed for about 1 minute

Add in the milk, yeast, salt, and vanilla

Turn the mixer to low, and then add in 2 cups of the flour

Attach the dough hook and then on medium speed, add the butter one piece at a time until smooth

Reduce the speed to low, and then add the rest of the flour until the dough sticks to the hook and pulls away from the sides of the bowl

It should be soft and moist, but not sticky

Turn the dough out on to a lightly floured surface and knead until smooth

Place in a mixing bowl coated with cooking spray and cover with a damp towel

Let rise until doubled in volume, about 1 hour

Punch down the dough and roll it out to about 1/2" thickness

With biscuit cutters, cut 3" circles with 1" holes

Preheat the oven to 400°F

On a baking sheet lined with parchment paper, place the doughnuts 1" apart

Cover with plastic wrap and let sit on top of the preheating oven until almost doubled in size, about 25 minutes

Bake until light in color, 5-8 minutes

Dip the hot doughnuts in the melted butter and then coat with the cinnamon-sugar mixture

Serve immediately; doughnuts are best eaten the same day they are made