Heat about 2 inches of frying oil to 375°F in a deep, heavy pan, leaving ample space above the oil.
Bring the water, butter, vanilla, sugar and salt to a boil. Take the pan off the heat and beat in all the flour.
Stir over the heat for 1–2 minutes until the dough gathers together. Transfer it to a mixing bowl and let it cool for 5 minutes.
Beat the eggs into the dough individually, adding the next only after the previous egg is incorporated.
Mix the coating sugar with cinnamon. Put the dough in a sturdy piping bag with the specified star tip.
Pipe 6-inch strips carefully into the oil, cutting the ends with scissors. Fry in small batches for about 4 minutes on each side, checking that the centers cook through.
Drain the churros briefly, toss them in the cinnamon mixture while warm, and serve.