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Preheat oven to 375°; to make the cookies—in a bowl combine the flour, baking powder, salt, allspice, cinnamon, and ginger; set aside.
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In another bowl, cream the butter and sugar until evenly combined.
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Add in the eggs, pumpkin, and vanilla; beat well.
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Add the dry ingredients and mix thoroughly; stir in the walnuts.
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Drop by rounded teaspoonfuls onto ungreased cookie sheets, leaving several inches between for expansion.
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*The batter will seem extremely soft compared with most cookie doughs, but it will firm up during baking.
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Bake for 12 minutes; cool the cookies on the sheets for 10-12 minutes, then removed to a wire rack to cool completely.
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To make the frosting: add sugar, milk, and vanilla to a bowl; set aside.
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In a small saucepan over medium-high heat, cook the butter until lightly browned, about 3-5 minutes, stirring frequently.
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Remove from the heat, add to the other ingredients and beat until smooth and creamy; cover until ready to use.
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When the cookies are completely cool, spread a generous amount of frosting on each cookie, and top with a walnut half.
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Let icing set before stacking the cookies or they will stick together.