Heat oil to 360°F:

Pour vegetable oil into a large skillet or heavy-bottomed saucepan to a depth of 1½ to 2 inches and heat over medium-high. Clip a candy or deep-fry thermometer to the side and heat oil to 360°F (180°C).

Mix cinnamon sugar coating:

Combine 1/2 cup (100 g) granulated sugar with 1-2 tsp ground cinnamon in a small bowl, whisking well; reserve for coating.

Make choux dough:

In a medium saucepan over medium-high heat, place 1/4 cup (57 g) cubed unsalted butter, 1 cup water, 1 Tbsp (12 g) granulated sugar, and a pinch of salt. Bring to a boil, then add 1 cup (120 g) all-purpose flour and stir vigorously with a wooden spoon over low heat until a soft dough forms. The dough may be lumpy with some flour not fully incorporated. Remove from heat and let cool about 5 minutes.

Beat dough with egg:

Move the dough to a stand mixer bowl (or a large bowl for a hand mixer). Working quickly, add 1 large egg and 1 tsp vanilla extract, then beat on medium-high speed for 30 seconds until the dough, egg, and vanilla are fully combined. Work fast to avoid scrambling the egg from the hot dough. Transfer the dough to a disposable piping bag fitted with a large open star tip.

Pipe and fry churro bites:

Slowly pipe the dough over the hot oil, using clean kitchen shears to cut off 1 to 1½-inch lengths, allowing the dough pieces to drop into the oil (mind the splashing). Fry 6-8 churro bites at a time, turning them occasionally with a slotted spoon, for 90 seconds to 2 minutes until golden brown and crisp. Move them to a paper-towel-lined plate to drain.

Coat and cool churros:

Continue frying the remaining churro bites in batches. While one batch is frying, coat the already-cooked bites in the cinnamon-sugar mixture and place them on a separate plate or baking sheet to cool. Churros are best eaten the same day but can be stored in a paper bag for up to a day; note they will be less crisp.