Combine water, butter, brown sugar, and salt in a saucepan over medium heat and bring to a simmer.

Add flour all at once and cook, stirring, until a soft, sticky dough ball forms and a starchy film coats the bottom of the pan, about 5 minutes.

Continue cooking for about 1 minute more, scraping the film off the bottom of the pan as you stir.

Transfer dough to a bowl and let cool for 10 minutes.

Add in vanilla extract and eggs, one at a time, mixing in using a spatula until it becomes a sticky dough.

Preheat the oven to 425 degrees F (220 degrees C) and line 2 baking sheets with silpat mats or parchment paper.

Transfer dough into a pastry bag fitted with a star tip and pipe out 3 to 4 inch ropes onto the prepared baking sheets.

Spray tops of the churros with vegetable oil spray, and then spray water generously all over the pan.

Bake until churros are puffed and the edges are browned, 20 to 25 minutes.

Turn off the oven and open the door for about 10 seconds to vent heat. Close the door, leaving it open about 8 to 12 inches, and let churros rest in the oven for 10 minutes.

Remove from the oven and brush churros lightly with melted butter, then toss to coat in cinnamon sugar.