In a medium saucepan over medium heat, combine the water, butter, brown sugar, and salt. Bring the mixture to a simmer. Once simmering, add the flour all at once. Stir constantly as you cook, until the dough forms a soft, sticky ball and a starchy film appears on the bottom of the pan—this should take about 5 minutes. Continue stirring for an additional minute, scraping up the film with your spoon as you go. Although the pan won’t be completely clean, the dough will absorb some of the starchy residue.

Transfer the dough into a mixing bowl and allow it to cool for approximately 10 minutes. When the dough has cooled slightly, add the vanilla extract. Then, add the eggs one at a time, mixing thoroughly with a spatula after each addition. Be patient if the mixture looks separated at first—it will gradually come together into a smooth, sticky dough.

Preheat your oven to 425°F (220°C) and line two baking sheets with silicone mats or parchment paper. Fit a pastry bag with a star tip (number 356 is ideal) and fill it with the churro dough. Pipe ropes of dough about 3 to 4 inches long onto the prepared sheets, leaving a few inches between each churro. Use the back of a knife to cut the dough cleanly at the end of each pipe.

Lightly spray the tops of the piped churros with vegetable oil spray, then generously mist the entire baking sheet with water. Bake in the preheated oven until the churros puff up and their edges turn golden brown, usually around 20 to 25 minutes. Once done, turn off the oven, open the door briefly to release some heat, then close it but leave it slightly ajar (about 8 to 12 inches). Let the churros rest inside the warm oven for 10 minutes.

Remove the churros from the oven. Working in batches of six, lightly brush each churro with melted butter. Place the churros into a bag filled with the cinnamon sugar mixture and gently toss to coat them evenly. Repeat this process until all churros are fully coated. Serve immediately and enjoy your delicious homemade churros!