Mix the water with the butter and salt over medium to high heat on your stove. If making the vegan churros, mix the water with olive oil and salt.
When it begins to boil, remove it from the heat.
Immediately mix in the flour until it's completely incorporated into the water mixture. If making the vegan version, you can skip to the shaping step. You can add water, if needed, to get a thick, workable dough.
For the choux dough version, add the egg, and mix well until all of the ingredients are completely combined. You will end up with a very thick batter.
Heat the olive oil in a pan over medium heat. You'll want to use enough oil for the churros to be able to float in the pan without touching the bottom.
To shape the churros, load the batter into a churros dispenser, heavy-duty piping/frosting bag, or a cookie press. I use an electric cookie press with a large star-shaped disk. If you are using a piping bag, use the biggest tip you can find. A star-shaped tip will make your churros look more authentic.
Pipe the churros onto a clean cloth and cut them to a size that will fit in your pot for deep frying.
When the oil has warmed to around 190ºC/375ºF, you can begin to add the piped dough into the hot oil.
Cook until the churros are golden brown.
Remove from the oil, and immediately sprinkle with granulated sugar.
Continue with the process until you have finished using up all of your dough.
To make the churros ahead of time, the dough can be dispensed onto a tray with parchment paper and frozen. They can then be stored in a freezer bag until you are ready to fry them. (You can fry them directly from frozen or defrost them first.)