For the doughnuts:
¼ cup granulated sugar
1 cup milk, heated to 115 degrees
1 tablespoon active dry yeast
1 teaspoon salt
2 teaspoon vanilla extract
2½ to 3½ cups unbleached flour, divided, plus more for kneading
1 stick butter (4 ounces), cut into 1" cubes
For the cinnamon-sugar coating:
1 stick butter (4 ounces), melted
1 cup granulated sugar + 2 tablespoons cinnamon, mixed together
1. 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.
2. 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.
3. Punch down the dough and roll it out to about ½" thickness. With biscuit cutters, cut out 3" circles with 1" holes. Preheat the oven to 400 degrees. 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.
4. Bake until light golden 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.