In a large bowl whisk together the flour, sugar, baking powder, salt, and cinnamon; set aside.
In a small saucepan bring together the milk, white vinegar, vanilla, egg, and butter.
Cook over medium-heat, stirring constantly with a whisk until the butter melts, about one minute. Remove from heat.
Add the milk mixture into the flour mixture; mix until combined.
Using a large tablespoon, spoon the doughnut mixture into an ungreased doughnut pan.
Bake for 12 to 14 minutes, or until a toothpick inserted comes out clean.
Remove and immediately invert the pan onto a cooling rack.
While the doughnuts are cooling, begin to work on the glaze.
Glaze
Place all the ingredients in a small saucepan and cook over medium-heat, stirring constantly just until combined and no lumps appear.
Set the saucepan over a bowl filled with hot water.
Dip the doughnuts into the glaze, one at a time, and set on a rack, glaze side up. Put something underneath the rack because some of that glaze will drip off.