In a medium saucepan over medium heat, combine 1 cup of water, 1/2 cup of butter, 1/4 cup of brown sugar, and a pinch of salt. Stir until the mixture is simmering, and the butter has melted completely.

Once the mixture is simmering, add 1 cup of all-purpose flour all at once, stirring vigorously with a wooden spoon for about 5 minutes, until the dough is smooth and pulls away from the sides of the pan.

Transfer the dough into a mixing bowl and let it cool for about 10 minutes.

After the dough has cooled slightly, stir in 1 teaspoon of vanilla extract until well combined.

Incorporate 2 large eggs into the dough, one at a time, mixing thoroughly after each addition.

Line two baking sheets with parchment paper and fill a piping bag with the churro dough.

Pipe the dough onto the baking sheets, creating 3-4 inch long ropes.

Lightly spray the tops of the piped churros with cooking oil and mist the baking sheets with water.

Place the baking sheets in the preheated oven and bake for 20-25 minutes, or until the churros are puffed up and golden brown.

Once baked, turn off the oven and leave the churros inside with the door slightly ajar for about 10 minutes.

Remove the churros from the oven and brush each one with melted butter while they're still warm, then roll them in a mixture of cinnamon and sugar.

Serve them warm, and consider offering some dipping sauces like chocolate or caramel.