Preheat the oven to 350 degrees F. and line two baking sheets with parchment paper and set aside.
Prepare the flax egg in a small bowl by combining the 2 tablespoons ground flaxseed with the 4 tablespoons water, stir, and set aside and allow it to thicken.
Place half of the rolled oats (1 cup) into a food processor and pulse for about 10 seconds until it resembles a coarse meal. Pour the processed oatmeal and reserved 1 cup of oatmeal in a medium bowl and mix in the 2 cups all-purpose flour, ½ teaspoon baking powder, ½ teaspoon baking soda, ½ teaspoon salt, 2 teaspoons ground cinnamon, and ¼ teaspoon ground nutmeg.
Using an electric mixer, mix together the 1 cup butter, 1 cup brown sugar and ½ cup granulated sugar. Add in the prepared flax egg (or 2 large eggs), 1 teaspoon vanilla extract, and then gradually add in the flour mixture until combined.
Using a scoop, place the dough on the baking sheets about 2-inches apart, if you do not have a scoop then just hand roll the dough and place on the baking sheet.
Bake for 10 to 15 minutes until the bottoms begin to brown, if you prefer a softer cookie, then you will want to remove them when the bottoms are golden brown. If you like a more crispy cookie allow them to bake longer. Cool on the baking sheet for 5 minutes and then transfer to a wire rack to cool completely.
Make the icing by mixing together the 2 cups powdered sugar and 3 tablespoons non-dairy milk together adding the milk in a little at a time until you get your desired consistency, but it should be a little thin.
Dip the tops of each cookie into the glaze, allowing the excess to drip off. Place back on the rack and allow the glaze to set. Store in an airtight container at room temperature.