Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius) and line a large baking sheet with parchment paper.

Prepare the flax egg by combining ground flaxseed and water in a small bowl; allow to sit for 5 minutes until thickened.

In a medium bowl, whisk together the flour, oats, baking soda, cinnamon, ginger, and salt.

In a large mixing bowl, whisk the melted coconut oil, brown sugar, prepared flax egg, and vanilla extract until well combined.

Gradually fold the dry ingredients into the wet mixture until no dry streaks remain.

Stir in the grated carrots, chopped walnuts, and raisins until evenly distributed.

Scoop rounded tablespoons of dough onto the prepared baking sheet, spaced 2 inches apart.

Bake for 10 to 12 minutes until the edges are firm and lightly browned.

Cool the cookies on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.

For the frosting, whisk together softened vegan cream cheese and powdered sugar until smooth, then drizzle over the cooled cookies.