a piping bag fitted with a star tip; a deep-frying thermometer
Melt the butter in a small saucepan over low heat. Add the salt, 1/4 cup sugar and 2 1/4 cups water to the butter and remove from the heat. Stir in flour with a wooden spoon until the mixture is smooth and a dough is formed. Transfer to a stand mixer fitted with the paddle attachment, then add the eggs one at a time, beating on medium speed, followed by the vanilla.
Transfer the mixture to a piping bag fitted with a star tip and set aside. Cut parchment paper into 5-inch squares and place them on a sheet tray. Pipe a cookie-sized round of the churro batter onto each parchment square, then freeze for 1 hour.
Heat the vegetable oil in a Dutch oven to 350 degrees F. Fit a rack inside a sheet tray.
Combine the cinnamon with the remaining 1/4 cup sugar in a small bowl. Carefully slide half the frozen churros off the parchment paper into the hot oil and cook until golden brown and crisp, 6 to 8 minutes, then transfer to the rack. Sprinkle each churro with the cinnamon-sugar mixture immediately after removing them from the fryer.
Top each churro with a scoop of ice cream and a drizzle of dulce de leche, if using.