With the peels still on, slice each banana in half lengthwise. In a small bowl, mix the granulated sugar with 1/2 teaspoon of the cinnamon, then sprinkle evenly over the cut sides of the bananas.
Line the air fryer basket with perforated parchment paper and preheat the air fryer to 400 degrees F (air fryer setting). Place the banana halves cut-side up in a single layer and air fry for 6–8 minutes, until the tops are bubbling and caramelized.
Meanwhile, melt the butter in a skillet over medium-low heat. Stir in the brown sugar, the remaining 1/2 teaspoon cinnamon, and the salt, and cook for 1–2 minutes, until the sugar dissolves.
Remove the skillet from the heat, carefully pour in the rum and vanilla, then return it to the heat. Simmer for 2–3 minutes, stirring, until the sauce is glossy and slightly thickened. (Never pour rum from the bottle over an open burner; keep a lid nearby in case of flare-ups.)
Slip the warm bananas out of their peels and into serving bowls. Spoon the warm sauce over the top, add vanilla ice cream or whipped cream, and serve immediately.