Stir together cinnamon and sugar in a small bowl; set aside. Bring milk, butter, and salt to a boil in a small saucepan over medium heat. Add flour, and cook, stirring, until mixture forms a ball and pulls away from sides of pan, about 30 seconds. Remove from heat, and let cool 3 minutes. Add eggs, and stir until batter is smooth. Spoon mixture into a pastry bag fitted with a large open-star tip (such as an Ateco #828).
Heat 4 to 5 inches oil in a large Dutch oven until it registers 330 degrees on a deep-fry thermometer. Holding pastry bag a few inches above the oil, squeeze out batter, snipping off 4-inch lengths with a knife or kitchen shears. Fry 6 to 8 churros at a time, turning once, until deep golden brown all over, about 5 minutes; transfer to a paper-towel-lined plate to drain.
Roll churros in cinnamon sugar. Serve immediately with hot chocolate.