In a medium saucepan combine the water, the 1/3 cup butter, the brown sugar, and salt. Bring to boiling over medium heat. Add flour all at once, stirring vigorously with a wooden spoon. Cook and stir until mixture forms a ball and pulls away from the side of the pan. Remove from heat. Cool for 10 minutes. Add egg and vanilla, beating well with a wooden spoon.
Preheat oven to 300 degrees F. Transfer mixture to a decorating bag fitted with a large open star tip. Line a baking sheet with waxed paper. Pipe 4x1-inch logs onto prepared baking sheet.
In a deep saucepan heat 3 inches of oil to 375 degrees F. Fry logs, a few at a time, in hot oil about 2 minutes or until golden brown on both sides, turning once. Drain on paper towels. Keep warm in oven while cooking remaining churros.
In a medium bowl combine the 1/4 cup granulated sugar and the 3/4 teaspoon cinnamon. Roll warm churros in cinnamon-sugar mixture to coat.