Preheat the oven to 350° F. Line two baking sheets with parchment paper. In a mixing bowl, beat together the butter, brown sugar, and vanilla until light and fluffy, about 3-5 minutes. Add the egg, mixing to combine. Next, add the pumpkin butter and molasses and mix to combine. Add the flour, baking soda, cinnamon, and salt, beating until combined. Fold in the candied ginger pieces. Let the dough rest for 10 minutes at room temperature. Roll the dough into tablespoon-size balls (if the dough is too sticky add 2-4 additional tablespoons flour). Place on the prepared baking sheet, spacing the cookies 2 inches apart. Bake for 8-10 minutes or until the cookies are just starting to set around the edges, the centers should be doughy. Tap the pan 2-3 times on the counter to set the cookies. Let cool on the pan.Meanwhile, make the icing. Add the butter to a pot set over medium heat. Allow the butter to brown lightly until it smells toasted, about 2-3 minutes. Remove from the heat. Whisk in the powdered sugar, 2 tablespoons milk, and a pinch of salt. Immediately spread the icing over the cookies, it will set quickly. Sprinkle with nutmeg or cinnamon, if desired. Store in an airtight container for up to 5 days.