In a medium saucepan, combine 1 cup (240 grams) water, butter, 1 tablespoon (12 grams) sugar, salt, and vanilla; bring to a boil over medium-high heat. Remove from heat; add flour, stirring with a wooden spoon until combined. Cook over medium-high heat, stirring constantly, until smooth and a skin forms on bottom of pan, 1 to 2 minutes.

Transfer mixture to the bowl of a stand mixer fitted with the paddleattachment, and beat at low speed until cooled slightly, about 1 minute. Add eggs, one at a time, beating until combined after each addition and stopping to scrape sides of bowl. (Batter will appear broken but will come back together as eggs are incorporated.) Add egg white; beat until a smooth, glossy dough forms. (Dough should pass the “V” test [see Notes] and will hold a slight peak when pinched between fingers.) Cover and refrigerate for 30 minutes to 1 hour.

In a large Dutch oven or a countertop fryer, pour oil to a depth of 2 inches, and heat over medium heat until a deep-fry thermometer registers 360°F (182°C) to 365°F (185°C). (See Notes.)

Line a rimmed baking sheet with paper towels.

In a large shallow bowl, stir together cinnamon and remaining ¾ cup (150 grams) sugar.

Place dough in a large pastry bag fitted with a 3/8-inch closed star piping tip (Ateco #844). Applying even pressure, carefully pipe dough in batches (2 or 3 at a time) in a straight line about 6 inches in length into hot oil. Using kitchen shears, cut dough at the end closest to the piping tip when it gets to desired length. (If dough is not staying straight, use tongs to help straighten dough immediately after piping, if desired.) Fry until puffed and golden brown, 1½ to 2 minutes per side. Remove from oil, and let drain on prepared pan. Toss hot churros in cinnamon sugar. Place on serving platter. Serve with Cinnamon Chocolate Sauce.