Preheat oven to 400F with the oven rack positioned in the middle of the oven. Flip a muffin pan upside down and place it on a baking sheet. Generously spray 8 of the muffin cups with non-stick cooking spray or grease with butter.
In a small bowl, combine sugar and cinnamon, set aside. In another bowl, whisk the egg and water and set aside.
Unfold pastry sheets on a lightly floured surface. Roll the pastry sheets into 14×10-inch rectangles (depending on your work space, you might have to do this one at a time). Lightly brush the tops of the pastry sheets with some of the egg mixture, and sprinkle half of the sugar and cinnamon on top.
Fold over one half over the other, egg side down so the cinnamon is sandwiched inside.
With a rolling pin, roll out the dough to press down the cinnamon mixture into the dough and add about 2-inches to the length of the dough. Cut ten thin strips of dough lengthwise with a pizza cutter. Cut two of the strips into 2-inch small strips for the bottom of the cups.
Place two of the strips over the top of eight muffin cups. For the sides, cover each cup, beginning at the bottom, with a thin cinnamon strip, spiraling up, until the entire muffin cup is covered. Using your hands, press the dough down against the muffin tip to give it a firm shape. Brush dough with egg wash and repeat with remaining dough.
Bake for 22 minutes until the tops and edges are golden and crispy to the touch.
Remove from oven and lightly brush melted butter over the cups and sprinkle with remaining cinnamon mixture. Allow churro cups to cool down to room temperature.
Once churro cups have cooled, remove from muffin pan, scoop a generous scoop of Dryer’s Slow Churned Caramel Delight Ice Cream into the cup and drizzle with a little caramel syrup.