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 the 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 the saucepan is full.
Cook, turning occasionally, until golden brown, about 5 minutes.
Transfer pastries to a paper bag to drain, about 30 seconds.