Preheat oven to 350 degrees. Line a large baking sheet (or two) with parchment paper.

To make the cookies, in a medium bowl, combine the flour, cinnamon, baking powder, baking soda, nutmeg, cloves, and salt. Mix well and set aside.

In a mixer fitted with the paddle attachment, beat together the butter, granulated sugar, and brown sugar until blended. Scrape down the sides of the bowl. Beat in the egg. Beat in the pumpkin puree and vanilla. Scrape down the sides of the bowl.

Beat in the flour mixture until blended.

Using a small cookie scoop or heaping tablespoon, drop the dough onto the prepared pan(s), about 2 inches apart.

Bake for 10 to 12 minutes, until a wooden pick inserted into the center of one cookie comes out clean or with little moist bits clinging to it. Cool the cookies on the pan(s), on a wire rack. Cool completely before frosting.

To make the frosting, using a mixer fitted with the paddle attachment, beat the butter until smooth and creamy. Beat in the maple syrup and vanilla. Beat in the powdered sugar 1/2 cup at a time and stop at 1 cup. Beat in 1 teaspoon of the milk.

If the frosting seems too thin, beat in the remaining 1/2 cup of powdered sugar. If it seems too thick, beat in the remaining milk.

Spread the frosting on cookies and serve. Store leftovers in an airtight container.