How to Make It
Combine 2 1/4 cups water, butter, vanilla bean seeds, orange zest, salt, and 1/4 teaspoon cinnamon in a large saucepan over medium; bring to a simmer, stirring occasionally. As soon as the mixture begins to simmer, add flour all at once, and quickly stir with a heatproof spatula or wooden spoon until completely incorporated, about 20 seconds.
Immediately transfer mixture to the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed 3 minutes. Add egg, and beat on medium speed until completely incorporated, about 20 seconds. Transfer batter to a double-lined pastry bag (place 1 bag inside another) fitted with a 1/3-inch open star tip.
Pipe 14 (15-inch-long) straight ropes of batter onto 2 parchment paper–lined baking sheets. Let dry until a skin forms and batter is no longer sticky around top and sides of rope, about 30 minutes. Bring ends of 1 rope together to form a teardrop shape, lightly pressing ends together to secure. Repeat with remaining ropes. Chill teardrop shapes at least 30 minutes or up to 2 hours. (Alternatively, freeze on baking sheets until hardened. Transfer to a large ziplock plastic freezer bag, and freeze up to 1 month.)
Stir together sugar and remaining 1 tablespoon cinnamon in a large bowl; set aside. Pour oil to a depth of 2 1/2 inches in a large Dutch oven. Heat over medium-high to 375°F. Working in batches, fry chilled or frozen churros, turning occasionally, until light golden brown with a crispy outside and custardy center, 2 minutes and 30 seconds to 3 minutes and 30 seconds. Toss hot churros in cinnamon-sugar, and serve immediately with cajeta or sweetened condensed milk for dipping.