Cover two baking sheets with parchment paper; set aside.
In a medium saucepan, stir together 1 cup water, brown sugar and salt.
Add butter and place over medium-high heat.
Heat until butter is melted and mixture starts to boil.
Remove from heat and add flour, stirring with a wooden spoon. Mixture will clump and pull away from the sides of the pan. Mix and mash with a wooden spoon until no streaks of flour can be seen.
In a small bowl, combine eggs and vanilla. Scramble mixture with a fork and then add to the dough ball in the saucepan.
Stir and mash, breaking up the dough until loosened. Stir well until eggs are incorporated and the mixture has the appearance of mashed potatoes.
Transfer dough to a piping bag fitted with Ateco decorator tip #867. You could also pipe the dough in a zip-top bag with the corner snipped to make churros with no ridges.
Pipe dough into long, thin lengths on the parchment-covered pans. Use a pair of kitchen scissors to cut the end of the dough from the piping tip (Image 1). Leave about 2 inches of space between the churros.
Bake for 10-12 minutes or until slightly puffed. Turn oven to broiler setting and watch carefully as churros toast and turn deep golden brown. Remove from oven and let cool slightly.
Transfer to a wire cooling rack. If you're baking the churros one pan at a time, be sure to return the oven setting to 425°F before putting in the next pan.
Combine sugar and cinnamon and pour onto a long dish.