Step 1: In a medium saucepan, combine 1 cup water, 1/4 cup unsalted butter, 2 tablespoons granulated sugar, and 1/2 teaspoon salt. Bring the mixture to a rolling boil over medium-high heat, ensuring the butter is fully melted.
Step 2: Remove the saucepan from the heat. Add 1 cup all-purpose flour all at once. Stir vigorously with a wooden spoon or spatula until the mixture forms a ball and pulls away from the sides of the pan. Return the pan to low heat and continue to stir for 1-2 minutes, pressing the dough against the bottom of the pan to dry it out slightly.
Step 3: Transfer the dough to a large mixing bowl. Let it cool for about 5 minutes. Add 1 large egg and beat with an electric mixer on medium speed (or vigorously by hand) until the egg is fully incorporated and the dough is smooth and uniform.
Step 4: Prepare your piping bag by fitting it with a large star tip (such as Wilton 1M or 21). Spoon the churro dough into the prepared piping bag.
Step 5: In a deep pot or Dutch oven, pour 3-4 cups of vegetable oil. Heat the oil to 350-375°F (175-190°C). Use a candy thermometer to ensure accurate temperature.
Step 6: Carefully pipe strips of dough, 4-6 inches long, directly into the hot oil. Use kitchen shears or a knife to cut the dough from the tip. Fry 3-4 churros at a time to avoid overcrowding and maintain oil temperature.
Step 7: Fry the churros for 2-3 minutes per side, or until they are golden brown and cooked through. Use tongs to carefully flip them. Once fried, remove the churros from the oil and place them on a parchment-lined baking sheet to drain excess oil.
Step 8: In a shallow dish, combine 1/2 cup granulated sugar and 1 tablespoon ground cinnamon. While the churros are still warm, roll them generously in the cinnamon sugar mixture until fully coated.
Step 9: Serve the churros immediately with your favorite dipping sauce, such as chocolate or dulce de leche.