To make the chocolate sauce: Pour half the milk into a small saucepan and bring to a gentle simmer. Add the chocolate and sugar and stir until the chocolate is melted. Mix together the remaining milk and cornstarch, then add to the pan of chocolate and stir constantly while the mixture simmers for about 2 minutes, it will become quite thick. Turn off heat, leave in pan and cover until ready to use. This can be made ahead, stored in a glass jar and reheated in the microwave for about 30 seconds.
To make the Churros: In a medium saucepan, bring the water and butter to a boil. Add the flour and salt and stir until the it thickens to a paste. Continue to stir until it forms a smooth ball. Put the dough into a stand mixer fitted with a paddle attachment and mix on low for about 20 seconds. Add one egg and mix until it is fully incorporated. Scrape down the bowl, and add the second egg, mix until dough is smooth. Scrape the bowl and add the final egg. When the dough is smooth, place it in a piping bag, fitted with a large star tip.
Heat the oil to 380°F on a candy thermometer. Squeeze the dough out into the hot oil in any shape you like. Be sure not to over crowd them. I had fun creating odd shapes, but you can make more traditional sticks as well. Cut the dough off with a knife when they are the length you desire. Fry until golden brown, then flip and fry the other side. About 4 minutes total. Use a slotted spoon to retrieve them from the oil and place them on paper towel to drain. Repeat as many times as it takes to use up all the dough.
Dredge in sugar mixed with the cinnamon. Dip the Churros into the Hot Chocolate Sauce and/or Cajeta.