Preheat the oven to 350°F. Line a baking sheet with parchment paper and set aside. Prepare the vegan egg replacer according to package directions. Set aside.

In a stand mixer or medium mixing bowl with a hand mixer on medium speed, cream together the butter, light brown sugar, and granulated sugar. Add the egg replacer and vanilla extract and continue mixing just until incorporated.

Add the all-purpose flour, cinnamon, baking soda, nutmeg, salt, and cloves. Mix just until incorporated; the dough will be thick.

Add the shredded carrots and walnuts. On low speed, mix until carrots and walnuts are well distributed throughout the dough.

Transfer the mixing bowl to the refrigerator and let the dough chill for 30 minutes.

Roll 2 heaping tablespoons of cookie dough into a ball and place on the prepared baking sheet. Repeat with the remaining dough, spacing the rolled balls 2 inches apart. If working in batches, return any excess cookie dough to the refrigerator until you are ready to use it again.

Bake for 12 minutes and allow cookies to cool completely on the baking sheet.

For the Frosting:

In a stand mixer or medium mixing bowl with a hand mixer on medium speed, cream together the cream cheese and butter. Add the vanilla extract and continue mixing to incorporate.

Add the powdered sugar and continue mixing until smooth.

Using a small spatula or spoon, spread frosting over the tops of the cooled cookies. Sprinkle with chopped walnuts and serve.