Grate the carrots and set aside. Add the room-temperature vegan butter, pure vanilla extract and brown sugar to the bowl of a stand mixer (or use a bowl with a hand-held mixer) Mix until combined- the consistency should be creamy and fluffy. In a medium mixing bowl, whisk together the all-purpose flour, rolled oats, cinnamon, nutmeg, baking soda and salt.
Add the dry ingredients to the butter mixture and process until combined. Add the grated carrots and walnuts and mix until combined.
Transfer the dough to a covered bowl and chill in the refrigerator for 30 minutes.
Preheat the oven to 325°F and line a baking sheet with parchment paper. Take 1 1/2 tablespoons of dough and roll it into a ball. Slightly flatten the top out with your hands and place on the baking sheet. Place the cookies 2 inches apart because they will expand as they cook.
Bake for 14 minutes or until the sides are golden brown. The cookies will seem soft when you take them out of the oven, but they will firm up as they cool. Let cool on the baking sheet for 5 minutes and then transfer them to a wire rack so they can cool completely.
While the cookies are cooling, make the vegan cream cheese frosting. Beat the vegan cream cheese, dairy-free butter and powdered sugar together until smooth and creamy. Spread a generous amount of frosting on the flat side of one cookie (you can also pipe it on) and then sandwich it with another cookie. Repeat until all the cookies have been sandwiched together.