Combine 1 cup water, 1/2 cup unsalted butter cut into pieces, 1 tbsp granulated sugar, and 1/2 tsp fine sea salt in a medium saucepan over medium-high heat, bringing to a rolling boil.

Reduce heat to medium, add 1 cup all-purpose flour all at once, and stir vigorously with a wooden spoon until a smooth dough ball forms and a thin white film coats the bottom of the pan, about 1 to 2 minutes.

Remove the pan from heat and let the dough cool for 5 minutes until warm to the touch but not scorching hot.

Add 1 large room-temperature egg and 1 tsp vanilla extract to the dough, beating vigorously with a wooden spoon or electric mixer until fully incorporated and smooth.

Transfer the warm dough into a heavy-duty piping bag fitted with a 1/2-inch open star tip.

In a shallow bowl, whisk together 1/2 cup granulated sugar and 1 1/2 tsp ground cinnamon until thoroughly combined, then set aside near your frying station.

Pour 4 cups vegetable oil into a deep Dutch oven or heavy saucepan (oil should measure roughly 2 inches deep) and heat over medium-high heat until it reaches 360°F to 370°F on a candy thermometer.

Line a large baking sheet with two layers of paper towels and set it next to your stove.

Hold the piping bag directly over the hot oil, pipe a 4-inch to 5-inch length of dough, and cut the end cleanly with kitchen shears, letting the dough drop gently into the oil.

Pipe 3 to 4 churros per batch, avoiding overcrowding the pot so the oil temperature does not drop below 350°F.

Fry churros for 2 to 3 minutes per side, turning with metal tongs or a slotted spoon, until deep reddish golden-brown on all sides and crisp to the touch.

Use a slotted spoon to transfer fried churros to the paper towel-lined baking sheet for 15 seconds to drain excess oil.

Roll warm churros immediately in the cinnamon-sugar mixture until evenly coated, then serve hot.