Preheat oven to 350˚F. Grease and flour a doughnut pan.

In a large bowl, whisk together the flour, sugar, baking powder, salt, and cinnamon. Set aside.

In a small saucepan, combine milk, white vinegar, vanilla extract, egg, and softened butter.

Cook over medium heat, stirring constantly until butter melts and mixture is combined, about one minute. Remove from heat.

Add the wet ingredients to the dry ingredients and mix until just combined. Do not overmix.

Spoon the doughnut batter into the prepared doughnut pan, filling each cavity about two-thirds full.

Bake for 12 to 14 minutes, or until a toothpick inserted into the center comes out clean.

Remove from oven and immediately invert the pan onto a wire rack to release the doughnuts.

While the doughnuts are cooling, prepare the glaze.

In a small saucepan, combine milk, vanilla extract, and powdered sugar. Cook over medium-low heat, whisking constantly, until smooth and slightly thickened.

Once the doughnuts are cool enough to handle, dip each one into the glaze, letting excess drip off. Place on the wire rack.

Let the glaze set for a few minutes before serving.