Make the egg alternative: Mix ground flaxseed with water and set aside to thicken.
Pulse the oats 4-5 times using a blender or food processor to create a flour mix - some powder and some chopped oats.
Combine processed oats, flour, salt, baking soda, cinnamon, and nutmeg into a medium bowl and mix.
In a large bowl beat the butter, brown sugar, and vanilla together on medium-high speed until creamed, about 2 minutes. Mix in the flax mixture. Add dry ingredients to the wet ingredients and mix with a spoon until combined.
Cover bowl and chill in the fridge for 30 minutes.
Preheat oven to 350 degrees F and line baking sheets with parchment paper.
Scoop cookie dough, about 1 1/2 tablespoons of dough per cookie, and place on baking sheet about 2-3 inches apart. Bake 11-12 minutes, until golden brown. The center will still look soft.
Set cookies aside and allow to cool completely before adding the icing.