2 large eggs plus 1 large egg yolk
6 tablespoons plus 1 cup sugar
2 1/2 tablespoons orange juice
1 tablespoon finely grated orange peel
1 1/2 teaspoons vanilla extract
2 3/4 teaspoons ground cinnamon, divided
1 1/2 cups self-rising flour
Vegetable oil (for deep-frying)
Whisk eggs, egg yolk, 6 tablespoons sugar, orange juice, orange peel, vanilla, and 1/2 teaspoon cinnamon in large bowl. Let stand 5 minutes, then stir in flour. Mix 1 cup sugar and 2 1/4 teaspoons cinnamon in medium bowl.
Pour enough oil into heavy large saucepan to reach depth of 1 inch. Attach deep-fry thermometer to side of pan and heat oil to 355°F to 360°F over medium-high heat. Spoon batter into pastry bag fitted with 3/8-inch (medium) star tip. Working in batches, pipe several 3- to 4-inch-long ribbons of batter into hot oil. Fry until golden brown, turning occasionally and adjusting heat to maintain temperature, 1 1/2 to 2 minutes. Transfer to paper towels to drain.
Working in batches, add warm churros to bowl of cinnamon sugar and toss to coat.