Preheat the oven to 350°F. Line baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, cloves, and salt.
In a large bowl, beat the butter, granulated sugar, and brown sugar until creamy. Beat in the egg, then mix in the pumpkin purée and vanilla. Scrape the bowl as needed.
Add the dry ingredients in two additions. Mix on low or fold with a spatula just until combined. The dough will be soft and sticky.
Scoop approximately 2 tablespoons of dough per cookie onto the prepared sheets, leaving about 2 inches between the portions.
Bake one sheet at a time for 12–15 minutes, until the tops are set and spring back lightly when gently pressed. Repeat with the remaining dough, letting used baking sheets cool between batches.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack. Cool completely before frosting.
Beat the cream cheese and butter until smooth. Mix in the vanilla.
Gradually add the sifted confectioners' sugar, mixing on low at first. Beat just until creamy. If needed, refrigerate for 15–20 minutes to firm slightly.
Spread frosting over each cooled cookie. Sprinkle lightly with cinnamon if desired. Refrigerate the frosted cookies until serving.