In a large mixing bowl, combine the melted butter, brown sugar, molasses, salt, cinnamon, ginger, allspice, egg, baking powder, baking soda, and flour. Mix until a smooth dough forms.
Cover and refrigerate the dough for about 1 hour, until firm.
On a lightly floured surface, roll the dough to ¼ inch thick, flouring the rolling pin as needed.
Cut out shapes with a gingerbread man cookie cutter. Return scraps to the fridge between batches.
Line the air fryer basket with parchment paper or spray with cooking spray. Place cookies in a single layer, leaving space between them.
Air fry at 320°F for 5 to 7 minutes, until the edges just begin to brown.
Let cool in the basket for 2 to 3 minutes, then transfer to a wire rack to cool completely.
Decorate with icing once fully cool. Serve and enjoy!