Make the dough: Combine the butter with 1 cup water and the salt in a medium saucepan over medium heat; grate in the zest of 1 lime. Let the butter melt, then bring the mixture to a simmer and slowly whisk in the flour. Remove from the heat and keep whisking until the dough forms a ball and pulls away from the side of the pan. Whisk in the eggs, one at a time, mixing each in completely before adding the next.
Fry the churros: Fit a pastry bag with a medium open star tip. Prop up the bag in a glass and cut the bag over the edge. Fill with the dough and refrigerate 15 minutes. Meanwhile, fill a large pot with 2 1/2 inches of vegetable oil. Heat until a deep-fry thermometer registers 350 degrees F. Working in batches, slowly squeeze 3-inch-long strips of dough into the oil, breaking off the dough with your fingers or a knife. Working in batches, fry the churros, 5 to 6 minutes, flipping halfway through, until golden brown and crisp. Remove the churros with tongs and drain on a paper towel-lined baking sheet. Let cool slightly.
Coat the churros: Combine the sugar, cinnamon and the zest of the remaining lime in a small paper bag. Working in batches, add the churros to the bag and shake to coat in the cinnamon sugar. Serve warm.
To get authentic ridges, use a medium open star pastry tip. The opening should be about the width of your pinky finger.