Preheat oven to 375 degrees. In a medium bowl, whisk together flour, baking soda, baking powder, pumpkin-pie spice, and salt; set aside.
Using an electric mixer, beat together butter and sugar until light and fluffy. Add egg; beat until smooth. With mixer on low speed, alternately add flour mixture in two parts and pumpkin puree in one, beginning and ending with flour mixture; mix just until combined (do not overmix).
Drop dough by heaping tablespoons onto two baking sheets, about 1 1/2 inches apart. Bake until puffed and edges are golden, 15 to 20 minutes, rotating sheets once during baking. Immediately transfer cookies to wire racks, and cool completely.
When cookies have cooled, set them (still on rack) over a baking sheet or waxed paper. Place chocolate in a medium heatproof bowl set over (not in) a saucepan of simmering water; stir until almost melted. Remove from heat; stir until completely melted. Pour chocolate into a resealable plastic bag. Snip off a corner with scissors or a knife to make an 1/8-inch hole; pipe chocolate over cookies. Refrigerate until chocolate is firm, about 20 minutes.
Store cookies in layers divided by parchment paper in an airtight container up to three days. A resealable plastic bag works well for the simple piping in this recipe. Fill it with chocolate, seal while squeezing out air, and twist into a cone; then snip off the corner and pipe.