Baked Cinnamon Sugar Doughnuts

INGREDIENTS

For the doughnuts:

egg

1/4 cup

granulated sugar

1 cup

milk (heated to 115 degrees)

1 tbsp

active dry yeast

1 tsp

salt

2 tsp

vanilla extract

2 1/2

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)