Preheat the oven to 350F.
Add the butter, sugar, and brown sugar to a large bowl. Use a stand mixer or hand mixer to mix until well combined, about 2 minutes.
Add the egg, molasses and vanilla extract. Mix until combined, about 1 minute.
Add the flour, oats, cinnamon, ginger, baking powder, and salt. Mix on low until just combined, about 1 minute.
Use a large (1/4 cup) cookie scoop to scoop out the dough. Place up to 4 dough balls on a baking sheet lined with parchment paper.
Transfer to the oven and bake for 8-10 minutes, until the edges are just starting to brown. Let the cookies cool on the baking sheet for 10 minutes, then transfer to a cooling rack to cool completely.
For the icing, add the powdered sugar, milk, vanilla, and salt to a bowl. Whisk until combine. When the cookies are completely cooled, drizzle the icing over each one. Enjoy!