In a medium-sized pot over medium-high 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; bring the mixture to a rolling boil. As soon as the mixture boils, add in 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.
Using a rubber spatula, continue to mix pressing out any lumps, until the dough is smooth, another 30 or so seconds (don't over-mix dough). Let dough cool for 5 minutes and then add in the vanilla and one egg. Stir. 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 again, stir until incorporated. The dough is thick and gluey, but keep stirring and it will come together. Again, avoid over-mixing the dough. As soon as it's cohesive, stop stirring.
Transfer the mixture to a large piping bag fitted with an open star tip.
Preheat a deep fryer (or heavy bottomed large pot over medium heat) to 350 degrees F. Fill with oil -- about 3 cups (you may need to add more as you fry). I highly recommend a candy thermometer to ensure the oil heat stays consistent; this is the best way to tell if Churros are frying evenly.
While 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, Set aside. Set out a large plate lined with a few paper towels.
Pipe 3-4 Churros at a time into the deep fryer or pot. Pipe the dough and then use kitchen scissors to cut the dough from the star tip into the oil -- do this carefully to avoid oil splattering and burning your arm.
Allow the Churros to fry until they are golden brown, flipping about halfway through the cooking time for a total of about 2 minutes per side. I like to use a wire skimmer to remove the Churros. Transfer to paper towels to dry briefly (but don't wait long) and then transfer to prepared cinnamon-sugar mixture and roll to generously coat.
Repeat with remaining dough, being careful to not add more than 4 Churros to the pot at a time and watching the thermometer closely to ensure even/consistent 350 degree temperature.
Enjoy while hot with chocolate sauce and/or dulce de leche. Churros are best enjoyed right after being fried.