In a medium mixing bowl, combine the flour, baking soda, baking powder, cinnamon, cloves, and salt until well blended. In a large mixing bowl, use an electric mixer to beat the butter with the brown sugar and granulated sugar and well blended. Beat in the pumpkin puree, egg, and 1 teaspoon vanilla extract until completely blended. Gradually beat in the flour mixture.
Drop the batter by heaping tablespoons on a baking sheet with at least 2 inches between each cookie. Bake for about 16 minutes until the edges are firm and the middle isn’t mushy. Cool on the baking sheets for 2 minutes, then transfer the cookies with a spatula to a wire rack to cool completely.
While the cookies are baking and cooling, make the frosting. In a medium mixing bowl, use a mixer to blend together the confectioners’ sugar, milk, 1 tablespoon melted butter, 1 teaspoon vanilla extract, and pinch of salt until smooth. When the cookies have cooled completely, spread the frosting over the cookies. Let the frosting set for about 30 minutes at room temperature.
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Notes
Make sure to leave at least 2 inches between each cookie when you drop the batter on the baking sheet.
Cool the cookies completely before icing them.
The frosting will firm up after about 30 minutes, as long as your kitchen isn't too hot. You can chill them in the fridge for 20 minutes if it's too warm for the frosting to set.