For the churro dough: Heat 2 inches oil to 350 degrees F in a wide 3 1/2 to 4-quart heavy pot over high heat.
In a medium saucepan over medium-high heat, combine 1 cup water and the cinnamon stick and bring to a boil. Discard the cinnamon stick. Add the butter, sugar and salt and stir until the sugar dissolves and the butter melts. Turn off the heat. Using a wooden spoon, quickly stir in the flour until it is well combined and a shiny dough forms. Return the pan to medium-heat and cook the dough, stirring, until it starts to pull away from the sides of pan, about 2 minutes. Transfer the dough to a medium bowl and let cool, about 5 minutes. Beat in the eggs, 1 at a time, stirring vigorously after each addition, until the egg is completely incorporated (the dough will be sticky.)
Transfer the dough to a pastry bag and pipe 1-inch pieces, cutting the dough with scissors every inch, into the hot oil. Fry the churro bites, 6 at a time, turning occasionally, until golden brown, 3 to 4 minutes per batch. Drain on paper towels. Return the oil to 350 degrees F between batches.
For the cinnamon sugar: Combine the sugar and cinnamon in a shallow bowl.
For the Mexican chocolate sauce: Bring the heavy cream to a simmer over medium heat. Turn off the heat and add the chopped chocolate, stirring until completely dissolved. Serve warm as a dipping sauce for the churro bites.
Roll the warm churro bites in the cinnamon-sugar. Serve Mexican chocolate sauce alongside for dipping churro bites.