CHURRO DOUGH: In a small to medium-sized pot over medium heat, combine the 1/2 cup milk, 1/2 cup water, 1/4 teaspoon salt, 1 tablespoon white sugar, 1/4 teaspoon cinnamon, and 8 tablespoons butter. Mix until the butter is melted and bring the mixture to a rolling boil. As soon as the mixture boils, add the 1 and 1/4 cup flour. Reduce the heat to low and stir vigorously for 30 seconds to a minute and then remove from heat.
CHURRO DOUGH, CONT.: Using a rubber spatula, continue to mix (and press out any lumps) until the dough is smooth, another 30 seconds or so (don't overmix dough). Let the dough cool for 5 minutes and then stir in the vanilla and one egg. It will seem hard to mix and gluey, but mix until it comes together. Add another egg and stir until incorporated. Add the final egg and combine again. Keep stirring and it will come together. As soon as it's cohesive, stop stirring; don't overmix.
PIPING CHURRO DOUGH: Transfer the churro dough to a large canvas or cloth piping bag fitted with an open star tip. Line a large pan with parchment paper (don't skip the parchment!) and pipe long strips (10 inches) of the dough and roll each strip into a round. Alternatively, pipe the dough int rounds as it comes out of the pastry bag (just keep the rounds about 2-2.5 inches wide). Allow to rest for 15-20 minutes, or place in the freezer.
DEEP FRYER PREP: In the meantime, preheat a deep fryer (or heavy-bottomed pot over medium heat) to 350 degrees F. I recommend using a candy thermometer to monitor the oil temperature and keep it consistent.
CINNAMON-SUGAR COATING: While the oil is coming up to temperature, stir together the 3/4 cup white sugar, 1/4 cup brown sugar, and 1 tablespoon cinnamon in a small bowl and set aside. Set out a large plate lined with a few paper towels.
FRY CHURRO ROUNDS: Once oil reaches 350 degrees F, place 2-3 churro rounds at a time in the deep fryer or pot. Allow the churros to fry until they are golden brown, flipping about halfway through the cooking time (about 4-6 minutes total). Use a wire skimmer or fork to remove the churro rounds onto the plate lined with paper towels.
COAT IN CINNAMON SUGAR: Place the hot churros in the cinnamon-sugar mixture and coat evenly with the sugar.
ADD ICE CREAM: While still very warm, make the sandwiches by placing a small scoop of ice cream on one churro round, and top with another. Enjoy ice cream sandwiches immediately.
OPTIONAL CHOCOLATE GANACHE: If you want to add a chocolate ganache to the sandwich, prepare it by microwaving the heavy whipping cream in a microwave-safe bowl until very hot-- 45 seconds to a minute. Pour in the dark chocolate chips and stir until smooth and creamy. Dip the sandwich in the ganache or drizzle the chocolate over (or spread it on the churro rounds before filling with ice cream).
STORAGE: Fried churros don't store well or keep nicely; it's best if these churros are eaten immediately. The dough (unfried) can keep, covered and refrigerated, for up to a day.