Preheat the air fryer to 350°F for about 3 minutes.
Grease the basket lightly with the oil. Line with a parchment round if you have one.
Stir the brown sugar and cinnamon together in a small bowl.
Open the dough and separate the rolls.
Brush the tops with melted butter, then dip into the cinnamon‑sugar mixture.
Arrange rolls in the basket with space between each one.
Air fry 8–9 minutes, flipping halfway, until golden and the centers are set.
Whisk powdered sugar, milk, and vanilla until smooth. Adjust thickness to taste.
Rest rolls 2 minutes, then drizzle with the glaze or warm the packet icing and spread. Serve warm.