While the cookies cool, prepare the icing.
Place powdered sugar, rum, vanilla, and milk in a small bowl and mix with a fork to combine. If the icing is too thick to drizzle comfortably, add a 1/4 tsp of milk at a time. Its better on the thicker side than the thinner side, so only add as much milk just as necessary!
Once the cookies cool, drizzle them with icing, and let it set for 5-10 minutes.