Cut 5 pieces of kitchen parchment to fit into the bottom of the air fryer basket. In a small bowl, whisk together the eggs and vanilla; set aside. In a large saucepan, combine the butter, brown sugar, salt and 1 cup water. Cook over medium-high, stirring occasionally, until the butter has melted and the mixture begins to simmer.

Add the flour and, using a wooden spoon, mix vigorously until a smooth, lump-free dough has formed. Reduce to low and stir for 1 minute; the dough should pull away from the sides of the pan and leave a film on the bottom. Remove the pan from the heat. Stir in the egg mixture in 3 additions, stirring briskly with the wooden spoon after each; at first the eggs will resist incorporation but with continued stirring the mixture will become homogeneous. After all of the egg mixture is added, continue stirring until the dough is smooth, about 1 minute. Set aside and cool for about 20 minutes. Meanwhile, heat the air fryer to 350°F.

Transfer the dough to a large pastry bag fitted with a large open star pastry tip. Working one parchment sheet at at time, pipe the dough into thick 5-inch lengths at least 1 inch apart; you should be able to fit about 5 on each parchment sheet.

Remove the fryer basket from the air fryer. Lift the parchment sheet by its edges, place it in the basket and slide the basket back into the unit. Bake until the churros are golden brown, about 12 minutes. Meanwhile, in a pie plate, stir together the white sugar and cinnamon.

Using tongs, immediately transfer the churros to the cinnamon sugar. Toss and turn to coat all sides, then transfer to a wire rack set in a rimmed baking sheet. Continue piping, baking and sugaring churros until the remaining dough is used up. Serve the churros warm or at room temperature.