In a sizable saucepan set over medium heat, combine the butter, 2 tablespoons of sugar, and 1 cup of water. Bring this mixture to a boil, then stir in the vanilla extract. Once boiling, remove from the heat and add the flour and kosher salt. Stir the mixture vigorously with a wooden spoon until it thickens, approximately 30 seconds. Allow it to cool for about 10 minutes.
As the dough cools, combine the remaining 1 cup of sugar with the ground cinnamon on a rimmed baking sheet or a large plate.
Using a handheld mixer on medium speed, incorporate the eggs one at a time into the dough, beating thoroughly after each addition. Transfer this mixture into a piping bag fitted with a large open star tip.
Fill a large pot halfway with vegetable oil and heat it over medium-high heat until the oil reaches 375°F, as indicated by a deep-fry thermometer. Holding the piping bag a few inches above the oil, carefully pipe the churro dough into 6-inch ropes, cutting the dough with kitchen scissors as needed.
Fry the churros in batches, turning them occasionally, until they are golden brown, which should take about 4 to 5 minutes. Ensure the oil temperature remains at 375°F between batches. Using a slotted spoon or tongs, immediately roll the fried churros in the cinnamon sugar mixture, then place them on a wire rack to cool slightly.