Melt the butter in a 3.5 quart saucepan. Add the water, sugar, vanilla and salt. Stir and bring to a simmer over medium heat.
Once the water/sugar mixture starts to simmer, add the flour and turn the flame all the way down. Stir quickly until a mass of dough forms.
Transfer the hot dough to the bowl of your stand mixer. Allow the dough to cool. You can speed up the cooling process by mixing the dough on slow speed for 4 minutes using a paddle attachment.
Once the dough is just warm to the touch, add 1 egg at time. Let each egg incorporate fully into the dough before adding the next. Once the dough no longer looks wet and slimy add the next egg. Repeat this process until you've added and incorporated all the eggs.
Transfer the batter to a large piping bag fitted with a 1M star tip. Make sure not to over-fill the piping bag. You want to leave about 4 or 5 inches of space between the dough and the top of the piping bag. Twist the bag, secure it with a bread-tie and set aside.
Fill a pot or deep skillet with 2-3 inches of oil. Heat the oil to 325°F.
Hold the tip of the piping tip about 1-2 inches above the oil. Pipe 2-3 churros at a time into 4-inch long strips. Hold the pastry bag with one hand (after you've piped the churro) and use your free hand to snip the churro with a pair of clean kitchen scissors. Fry the churros for 1 minute on each side, or until they turn golden brown on all sides.
Transfer the churros to a plate lined with paper towels and fry the remaining batches.
Place the churros in a 170°F oven while you prepare the dulce de leche. You'll dust the churros in cinnamon/sugar after.