In a saucepan set over medium heat, place the water, butter, brown sugar, and salt. Warm the mixture until it reaches a gentle simmer.

Add the flour in a single addition once the liquid begins to simmer. Stir continuously as the mixture cooks, allowing it to gather into a soft, tacky dough ball. A pale starchy layer should begin to appear on the base of the pan after approximately 5 minutes.

Continue cooking the dough for 1 additional minute, using the spoon to loosen and work some of the starchy film back into the mixture. The bottom of the pan will not become fully clean, but the dough should absorb part of the residue.

Move the dough into a mixing bowl and allow it to cool for about 10 minutes.

Add the vanilla extract once the dough has cooled slightly.

Incorporate the eggs one at a time, stirring thoroughly with a spatula after each addition. The dough may appear broken at first, but continued mixing will bring it together into a glossy, sticky consistency.

Preheat the oven to 425°F / 220°C. Line two baking sheets with silicone baking mats or parchment paper.

Spoon the prepared dough into a pastry bag fitted with a star tip, ideally a number 356 tip.

Pipe 3- to 4-inch lengths of dough onto the lined baking sheets, leaving several inches of space between each piece. Use the back of a knife to cleanly separate the dough at the end of each piped strip.

Lightly coat the tops of the piped churros with vegetable oil spray.

Mist the entire baking sheet generously with water.

Bake the churros in the preheated oven for 20 to 25 minutes, or until they are puffed and the edges have turned golden brown.

Turn off the oven, briefly open the door to let some heat escape, then close it again while leaving it slightly open, about 8 to 12 inches. Let the churros remain in the warm oven for 10 minutes.

Remove the churros from the oven.

Working in batches of 6, brush the churros lightly with melted butter.

Place the buttered churros in a bag containing the cinnamon-sugar mixture, then toss gently until evenly coated.

Repeat the buttering and coating process until every churro is covered in cinnamon sugar.

Serve immediately while fresh.