Combine 1 cup water and 1 cup sugar in a saucepan over high heat. Bring to a boil; stir until sugar dissolves. Reduce heat to medium; add lime juice. Simmer until flavors combine, about 10 minutes. Remove from heat. Stir 1 teaspoon vanilla extract into the syrup; let cool.
Stir 2 cups water, 1 cup sugar, vegetable oil, and salt together in a pot over medium-high heat; bring to a boil. Reduce heat to medium. Add flour; stir vigorously with a wooden spoon until batter forms into a ball. Cook until batter is shiny, about 1 minute. Transfer to a bowl; allow to cool, about 5 minutes.
Flatten batter into the bottom of the bowl. Add eggs, 1 at a time, mixing well after each addition. Stir in remaining 1 teaspoon vanilla extract; mix until batter is creamy. Pour into a cloth piping bag fitted with a large star tip.
Heat corn oil in a deep saucepan over medium heat until bubbly; reduce heat to medium-low. Squeeze 2 inches of batter onto an oiled spatula and lower into the hot oil; repeat until saucepan is full. Cook, turning occasionally, until golden brown, about 5 minutes. Transfer pastries to a paper bag to drain, about 30 seconds.
Soak hot pastries in syrup; transfer to a plate to cool, about 5 minutes.
