In the bowl of a freestanding electric mixer fitted with a paddle attachment, combine 1 large egg and 1/4 cup granulated sugar. Beat on medium speed until well combined, about 1 minute. As you beat the sugar and egg, briefly whisk together 1 cup whole milk (warned to 115 (F)), 1 tablespoon active dry yeast, 1 teaspoon vanilla extract and 1 teaspoon salt, before adding it to the egg and sugar on low speed. In a separate bowl, whisk together 2 cups all purpose flour and 1 tablespoon ground cinnamon. Add to the mixture at low speed, about 1/2 cup at a time, and beat until the dough is thick and pulls away from the side of the bowl.
Once the dough is thick and pulls away from the bowl, stop the mixer and switch to the dough hook. Turn the mixer back on to medium speed and add 1/2 cup cubed unsalted butter one piece at a time. Continue beating until no large chunks of butter are left at the bottom of the bowl, around 3 to 5 minutes. Reduce speed to low and add additional flour one cup at a time (up to 1 1/2 cups) until the dough gathers around the hook and cleans the sides of the bowl. The dough should be soft and moist, but not overly sticky.
Turn the dough out onto a floured surface and knead gently until the dough no longer sticks to your hands. Transfer the kneaded dough to a lightly greased mixing bowl, turning the dough once over to lightly coat all of its surfaces. Cover with a damp tea towel and let rise in a warm spot until doubled in volume, about 1 hour.
Once the dough has risen, preheat the oven to 400 (F). Punch down the dough on a lightly floured surface and roll out to 1/2 inch thick. With a 3 1/2-inch donut cutter, cut out donuts and stamp out the holes. Place donuts at least 1 inch apart on a baking sheet lined with parchment paper or a Silpat mat. Cover loosely with plastic wrap and let sit in a warm spot until the donuts have doubled in size, about 20 minutes. When the donuts have proofed again, bake the donuts for 5 to 8 minutes in the preheated oven. The donuts will be a light golden brown — be careful not to overbake them! Transfer to a wire rack to cool for 30 minutes before glazing and eating immediately.