Cream butter and sugars until light and fluffy.
Mix in egg and vanilla until smooth.
Whisk dry ingredients and fold into butter mixture.
Shape dough into a log and chill for 30 minutes.
Slice dough, roll into balls, and press deep divots in centers.
Chill shaped cookies for 10-15 minutes.
Whisk pumpkin purée, sugar, egg, milk, flour, and spice for filling.
Fill each cookie center with pumpkin mixture.
Bake at 375°F for 3 minutes, then reduce to 350°F and bake 10 more minutes until set.
Cool completely on rack.
Whip cream with powdered sugar and pipe on cooled cookies.