Step 1: In a medium mixing bowl, combine the flour, brown sugar, baking powder, pumpkin spice, and kosher salt. Utilize a fork or a pair of table knives to incorporate the cubed cold butter into the flour mixture, mixing until it reaches a coarse, meal-like consistency.
Step 2: In a separate liquid measuring cup or bowl, whisk together the pumpkin puree and buttermilk until well blended. Gradually add this mixture to the flour mixture, stirring gently with a fork until just combined, forming a shaggy dough.
Step 3: Transfer the dough onto a lightly floured surface. Knead it gently by hand until no dry flour remains visible. Shape the dough into a square measuring approximately 4 inches on each side, then cut it into four equal squares. Brush the tops of these squares with the melted butter.
Step 4: Arrange the squares in a single layer in the air fryer basket. Cook at a temperature of 350°F (175°C), covering with foil for the last three minutes of cooking to prevent excessive browning. Continue cooking until the biscuits are golden and risen, approximately 10 to 12 minutes.