First, prepare the flax egg: In a small bowl, combine 1 tbsp ground flaxseed meal with 3 tbsp water. Stir well and let it sit for at least 5 minutes to thicken. It should have a gel-like consistency.

In a large mixing bowl, using an electric mixer, cream together the softened vegan butter, light brown sugar, and granulated sugar on medium speed until light and fluffy (about 2-3 minutes).

Add the prepared flax egg, 1/4 cup plant milk, and 1 tsp vanilla extract to the creamed butter mixture. Beat on low speed until just combined, scraping down the sides of the bowl as needed.

In a separate medium bowl, whisk together the all-purpose flour, old-fashioned rolled oats, baking soda, ground cinnamon, ground nutmeg, and salt.

Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix; overmixing can lead to tough cookies. The dough will be thick.

Cover the bowl with plastic wrap and chill the dough in the refrigerator for at least 30 minutes, or up to 1 hour. This helps prevent spreading and deepens the flavors.

Preheat your oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone baking mats.

Scoop rounded tablespoons of dough (about 1.5 inches in diameter) onto the prepared baking sheets, leaving about 2 inches between each cookie.

Bake for 10-12 minutes, or until the edges are lightly golden and the centers are set. The cookies will be soft when they first come out of the oven. Do not overbake.

Let the cookies cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely. They must be completely cool before icing.

While the cookies are cooling, prepare the icing. In a small bowl, whisk together the sifted powdered sugar, 2 tablespoons of plant milk, and optional 1/4 tsp vanilla extract until smooth. The consistency should be pourable but thick enough to coat a spoon. Add more plant milk, a teaspoon at a time, if the icing is too thick, or more powdered sugar if it's too thin.

Once the cookies are completely cool, dip the top of each cookie into the icing, or use a spoon to drizzle and spread the icing over the tops. Place the iced cookies back on the wire rack over parchment paper (to catch drips) and let the icing set for about 15-30 minutes, or until hardened.