Make the flax egg: Mix the ground flaxseed and water in a small bowl and set aside to thicken.
Using a blender or food processor, pulse the rolled oats 6-8 times until you have some oat flour and some chopped oats.
Add the processed oats to a medium sized bowl, along with the rest of the dry ingredients (flour, baking soda, salt and cinnamon). Mix to combine and set aside.
In a large bowl using a hand mixer, beat the softened vegan butter, brown sugar, molasses and vanilla until smooth and creamy. Mix in the flax egg as well. Now add the dry ingredients and mix with a large spoon until just combined. The dough will be thick and rather sticky. Cover the bowl, and chill the dough for about 30 minutes.
Preheat the oven to 350 degrees F and lightly grease a baking sheet (or line with parchment paper).
Scoop heaping tablespoons of dough (about 1.5 tablespoons each) and place 2-3 inches apart on the baking sheet. Press down slightly with your hands. If the dough sticks, get your hands a little bit wet first. Bake for 11-13 minutes, until golden brown. They will still look soft in the middle.
Allow the cookies to cool completely before icing.
Make the icing: Add the powdered sugar, vanilla and non-dairy milk to a small bowl and use a fork to whisk until combined and smooth. You want this icing to be very thick, so add more powdered sugar if needed, or a little bit more milk if it won't come together. It should be smooth but thick, not runny.
Once the cookies have cooled, dip the tops of each one lightly into the icing. The icing will take an hour or so to set completely, then you can stack them if desired. Enjoy!