Preheat the air fryer to 350°F for 3 minutes. Stir the granulated sugar and cinnamon together in a large bowl and set aside.
Unroll the pizza dough onto a lightly floured cutting board. Using a pizza cutter, cut into 1½-inch squares (about 30).
Brush the squares on both sides with about half of the melted butter.
Arrange in the air fryer basket in a single layer, leaving space between the squares. Air fry at 350°F for 8 minutes, shaking the basket at 4 minutes, until puffed and golden brown.
Transfer the hot squares immediately to the bowl of cinnamon sugar. Drizzle with the remaining melted butter and toss to coat. Coat while hot — the sugar will not stick to cooled dough.
Repeat with the remaining dough (2–3 batches total for a 6-quart basket).
For the icing, whisk the powdered sugar, vanilla, and salt with 1 tablespoon of milk. Add more milk a few drops at a time until it's thick but pourable. Serve warm with the icing for dipping.