Preheat oven to 425F. Roll out the puff pastry on a floured board into a rectangle, as thinly as possible. Stir the garlic juice into most of the milk and brush the pastry with half of it. Mix the paprika and Parmesan, and season with a little salt and cayenne. Sprinkle half of it over one half of the pastry. Fold the pastry and roll out as thinly as possible.
Repeat with the remaining garlic milk and Parmesan mixture and roll out to a rectangle not more than 1/4 inch thick. Brush with milk and cut into strips about 1/2 inch wide and 6 inches long.
Arrange the straws, at least an inch apart, on greased baking sheets and bake for 7 to 10 minutes or until well-risen and golden brown. Serve warm, piled onto each other, log cabin style.