Heat oven to 375° F.
Beat the butter, brown sugar, and granulated sugar in the bowl of an electric mixer fitted with a paddle attachment until smooth. Add the pumpkin, vanilla, and egg and beat until combined.
Combine the flour, baking powder, baking soda, pumpkin pie spice, and salt in a medium bowl. Slowly add the flour mixture to the sugar and butter and beat on medium-low speed until fully incorporated. Spoon heaping tablespoons of the mixture 2 inches apart onto parchment- or foil-lined baking sheets. Bake until puffed and cooked through, about 10 minutes. Let cool for 5 minutes.
Clean the mixer, then, as the cookies bake, beat the cream cheese, heavy cream, and confectioners' sugar until smooth and spreadable. Spread the flat sides of half the cooled cookies with the cream mixture. Top with the remaining cookies.
Tip: The cookies can be made up to 3 days in advance and kept in an airtight container. Whip the filling and assemble the sandwiches no more than 2 hours before serving.