First, make the sugar coating. In a medium bowl, combine 1/2 cup granulated sugar and 1 teaspoon ground cinnamon until fragrant and well combined. Set aside.
In a medium, heavy-bottomed sauce pot, combine 1 cup water, 7 tablespoons unsalted butter, 2 1/4 tablespoons dark brown sugar, and a pinch of kosher salt. Bring to a boil over medium heat, stirring until the butter has melted.
Once the butter has melted, take the pan off heat, and add 1 cup all-purpose flour and use a rubber spatula to beat into the mixture until it begins to form a smooth paste that pulls away from the sides of the pan. Return to heat and stir constantly for another minute or so.
After a minute, transfer the pastry mixture to the bowl of a freestanding electric mixer fitted with a paddle attachment. Beat the mixture on medium speed for a minute, before lowering the speed to its slowest setting and adding 2 eggs, only adding the second one when the first has been fully incorporated. Add 1/2 teaspoon pure vanilla extract.
Use a 1-tablespoon cookie dough scoop (in a pinch, use a tablespoon!) to portion out 1 tablespoon measures of the dough. Roll each portion into a ball with your hands.
Heat the vegetable oil in a pan or a deep-fryer until it reaches 350 (F). Drop a few of the churro balls into the oil and cook on each side for 2 to 3 minutes. When the churro balls become nice and golden, remove from heat and place on a wire rack for around 30 seconds or so, before transferring to the bowl containing the sugar coating (from the 1st step) and rolling the balls until they're nice and coated. Enjoy immediately.