Combine the butter and dark brown sugar in the bowl of a stand mixer fitted with the paddle attachment. Cream at medium speed for 2–3 minutes, or until light and fluffy. Scrape down the sides of the bowl, then beat in the egg.
Add the vanilla and molasses and mix on medium speed for 2 minutes.
In a separate medium bowl, whisk together flour, baking powder, ginger, cinnamon, baking soda, salt, nutmeg, and cloves.
Add the flour mixture to the butter mixture and process at low speed just until no streaks of flour remain.
Scrape the dough onto a sheet of plastic wrap. Wrap tightly and form into an even rectangle. Set in the fridge to chill for 1–2 hours or overnight. (If chilling for 24 hours, let the dough sit at room temperature for 1–2 hours before rolling).
Preheat the oven to 350˚F. Line two rimmed sheet pans with parchment paper.
Set a piece of parchment on a large work surface, then lightly flour the paper and rolling pin. Cut the dough in half (keep the remaining dough wrapped) and roll into a ¼ inch thick even rectangle, adding more flour as needed as you work. Use your preferred cookie cutters to cut out cookies, then place them on the prepared sheet pan. Reroll the scraps of dough and repeat process, adding more flour as needed, until no dough remains.
Bake for 8 minutes, or until set around the edges but soft in the middle. Let cool on sheet pans for 1 minute, then transfer to cooling racks to cool completely.
While the cookies cool, make the icing. Whisk the powdered sugar and milk until smooth and quite thick. You want the consistency to be stiff enough to pipe into even lines.
Once the cookies cool, transfer the icing to a piping bag and decorate each cookie as desired.