To make the churros, in a medium saucepan combine the butter, salt, vanilla extract, cinnamon stick and 2 1/4 cups water. Set over medium-high heat and bring to a boil. Remove from the heat, discard the cinnamon stick, and stir in the flour with a wooden spoon. Return to the stove, and cook, stirring constantly, until a smooth dough forms, about 5 minutes. Transfer dough to a bowl, and stir in the egg vigorously until the dough is smooth. Transfer to a large piping bag, fitted with a metal star tip. Set aside.

Combine the sugar and ground cinnamon in a shallow baking dish, until well combined. Line a plate or baking sheet with paper towels, set aside.

Fill a dutch oven halfway with oil, and set over medium-high heat. Using a deep-fry thermometer bring the oil to 400 degrees F. Fry the churros in batches by squeezing the dough through the piping bag into the hot oil, using scissors or your fingers to cut off about 4-inch churro pieces. Fry until golden brown all around, about 2 minutes. Using a slotted spoon, drain and transfer to the prepared plate to soak up the excess oil. While the churros are still warm roll around in the cinnamon sugar mixture. Serve the churros warm.

To make the chocolate sauce, in a medium saucepan combine the chocolate, butter, and cinnamon. Set over medium heat and cook, stirring often, until melted through. Stir in the sugar, and gradually stir in the evaporated milk. Bring to a boil, reduce heat, and simmer over low heat for 8 minutes, stirring constantly. Remove from heat, let cool slightly before serving with the churros for dipping.