Turn the oven on to 200°F if you wish to keep all of the waffles hot while you're cooking them. If you plan to serve them immediately after each batch is done cooking, you can skip this step.
Whisk together 1/4 cup granulated sugarr and 1/2 teaspoon cinnamon in a medium bowl to make the cinnamon-sugar coating. Set aside. Melt 1 Tablespoon unsalted butterr in a small, microwave-safe bowl on high power for 30 seconds. Set aside. (You may need to re-melt it again later for a few seconds if your kitchen is cool.)
Add 1 cup water, 1/4 cup unsalted butter, 1 Tablespoon granulated sugar, and 1/4 teaspoon salt to a medium saucepan over medium-high heat and bring to a boil.
Reduce the heat to medium-low and add 1 cup all-purpose flour. Stir the mixture with a rubber spatula, smoothing out any lumps of dry flour. Continue to cook and stir until the mixture is mostly smooth, with a thick, paste-like consistency, about 2-3 minutes.
Transfer the flour mixture to a mixing bowl and let it cool for 5-10 minutes.
In the meantime, crack 2 large eggs into a small bowl or measuring cup. Add 1/2 teaspoon vanilla extract and gently whisk them together with a fork. Now's also the time to get out your waffle maker and start pre-heating it.
Add the egg mixture to the bowl with the flour mixture. Beat together with a hand mixer until the egg is fully incorporated. Make sure to hold your mixer and the bowl firmly, as the batter will be very stiff (alternatively, you could use a stand mixer). Note that when you first add the eggs, the batter will intitially separate and look curdled. Keep mixing until the batter comes together. It will still be quite thick and pasty.
When the waffle maker is hot, brush or spray the surface of the waffle maker generously with canola oil. Spoon about 1-2 tablespoons of the batter into the center of each section of the waffle maker (eg mine makes two square waffles at once, so I put a total of two scoops of churro batter into the waffle maker at a time). Note that with this amount of batter, the churro waffles will be small and irregular-shaped. If you want to make full-sized waffles, you will need to use more batter and carefully spread the batter over the surface of the waffle maker, but then the recipe will make fewer waffles.
The waffles are done cooking when the waffle maker stops steaming and the waffles are golden brown on the outside. They make take a little longer to cook than the timer on your waffle maker indicates, if it has one.
While still hot, lightly brush the waffles on both sides with the melted butter, then sprinkle with the cinnamon-sugar mixture, tapping off any excess back into the bowl. Repeat with the remaining waffles.
Serve the waffles immediately with any desired garnishes, or keep them all hot in a 200°F oven while cooking the remainder of the waffles. If you need to reheat any leftover waffles, do so in a toaster oven, oven, or air-fryer as they will become soggy in the microwave.