Preheat your oven to 350°F (175°C) and grease two cookie sheets.
In a medium bowl, mix together the flour, cinnamon, baking powder, baking soda, nutmeg, cloves, and salt.
Cream the sugar and butter in a mixing bowl until fluffy; this should take about 2 to 3 minutes. Next, add in the pumpkin puree, egg, and vanilla extract, beating everything together until it's smooth and creamy.
Combine the flour mixture with the wet ingredients, stirring until fully mixed.
Using a tablespoon, scoop out dough portions onto the prepared cookie sheets. Gently flatten each portion slightly.
Bake in the preheated oven for 15 to 20 minutes, switching the positions of the racks halfway through. Once the centers are set, remove the cookies from the oven and let them cool on a wire rack until they reach room temperature (about 30 minutes).
While the cookies cool, make the icing: In a bowl, stir together confectioners' sugar, milk, melted butter, and vanilla extract until you have a smooth drizzling consistency. If needed, add more milk.
Once the cookies are cool, use a fork to drizzle the icing over them.
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