Heat the oven to 350°F (175°C) and line a rimmed baking sheet with parchment.
Melt the butter and let it cool for a few minutes. Whisk it in a large bowl with both sugars, then whisk in the eggs and vanilla until smooth.
Stir the baking soda, baking powder, salt and pumpkin pie spice into the flour. Add to the bowl and fold with a spatula until no dry flour remains.
Scoop with a 2 inch (5 cm) cookie scoop onto the sheet, 2 inches (5 cm) apart, about 8 per sheet.
Press a deep well into each ball without pressing through the bottom. Pinch closed any cracks in the walls.
Freeze the tray for 10 minutes.
Meanwhile, whisk the pumpkin puree, egg, brown sugar, milk and vanilla until smooth, then whisk in the cornstarch and pumpkin pie spice.
Fill each well to just below the rim, about 2 teaspoons each.
Bake 12 minutes, until the cookies have spread, the edges are golden brown and the filling looks set.
Cool on the sheet for 5 minutes, then on a wire rack until completely cool.
Repeat with the remaining dough, keeping it in the fridge between trays and freezing each tray for 10 minutes before filling.
Refrigerate the cooled cookies in an airtight container for up to 3 days.
Just before serving, top each cold cookie with whipped cream and a dusting of pumpkin pie spice.