In a bowl, stir together the butter, ground cinnamon, and brown sugar until smooth and spreadable. Set aside.
In a separate bowl, toss the diced apple with the salted caramel sauce until coated. Set aside.
On a lightly floured surface, roll the pizza dough into a 12 x 8-inch rectangle.
Spread the cinnamon butter over the dough, leaving a 1/2-inch border along one long edge. Scatter the caramel apples over the top.
Starting from the opposite long edge, roll the dough into a tight log and press the seam to seal. Cut into 6 rolls with a sharp serrated knife.
Line the air fryer basket with perforated parchment paper. Working in batches, arrange the rolls with space between them and air fry at 300 degrees F for 10 minutes.
Increase the temperature to 350 degrees F and cook for 10 to 12 minutes more, until deep golden brown and cooked through in the center.
Drizzle with powdered sugar icing or warm caramel sauce while still warm.