Position racks in the upper and lower thirds of the oven; preheat to 375 degrees F. Line 2 baking sheets with parchment paper; mist with cooking spray.
Whisk the flour, baking powder, baking soda, cinnamon, allspice and 1/4 teaspoon salt in a medium bowl. Whisk the brown sugar, pumpkin puree, vegetable oil, egg and 1/2 teaspoon vanilla in a large bowl; add the flour mixture and gently stir with a wooden spoon until just incorporated.
Drop tablespoonfuls of dough 2 inches apart on the prepared baking sheets. Bake 6 minutes, then remove from the oven and make an indentation in the center of each cookie with the back of a teaspoon. Return to the oven, switching the position of the pans, and bake until set, 4 to 6 more minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Make the filling: Beat the cream cheese and butter in a bowl with a mixer on medium-high speed until smooth. Add the confectioners' sugar, lemon juice, the remaining 1/2 teaspoon vanilla and a pinch of salt; beat until incorporated. Transfer to a pastry bag fitted with a small round tip; pipe into the indentations. Dust with cinnamon.