Homemade Hot Chocolate with Old-Fashioned Doughnut Sticks

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8 cups whole milk

16 ounces semisweet chocolate, chopped small or chips 

1/4 cup sugar

6 cinnamon sticks 

1 teaspoon vanilla extract 

Old-Fashioned Doughnut Sticks, recipe follows

Doughnuts:

Nonstick spray, for pan

1 1/2 cups all-purpose flour 

1/2 cup granulated sugar 

1 teaspoon baking powder 

1/4 teaspoon cinnamon 

1/4 teaspoon salt 

1/2 cup milk 

1 egg, lightly beaten 

1/2 teaspoon pure vanilla extract 

3 tablespoons unsalted butter, melted 

Glaze:

2 cups confectioners' sugar

2 to 3 tablespoons milk, more as needed