Bring the milk, chocolate, sugar, cinnamon and vanilla extract to a simmer in a medium saucepan. Whisk until completely melted and smooth.
Serve with the Old-Fashioned Doughnut Sticks.
For the dougnuts: Preheat the oven to 375 degrees F. Spray two 6-well eclair pans with nonstick spray.
In a large mixing bowl, whisk together the flour, sugar, baking powder, cinnamon and salt to combine. Add the milk, egg, vanilla and butter, and mix gently until combined.
Fill the wells of the prepared eclair pans about halfway with the batter.
Bake until the doughnuts spring back when touched and a toothpick comes out clean, 10 minutes. Transfer the pans to a rack set over a baking sheet; let cool for 3 to 5 minutes. Remove the doughnuts from the pans and let cool completely on the rack.
For the glaze: Whisk the confectioners' sugar and milk together until smooth.
Dunk the doughnuts in the glaze to coat completely. Transfer to a rack and let set for 10 minutes before serving.