Preheat the oven to 375°F. Line a baking sheet(s) with parchment paper. Set aside.
In a large bowl of a stand mixer, beat the butter with both sugars until light and creamy. Beat in the eggs, one at a time and beating well after each addition. Scrape down the sides as needed. Beat in the vanilla.
With the mixer on low speed, beat in the flour, baking soda, cinnamon, ginger, salt and nutmeg. Beat in the carrots, oats or quinoa flakes, and nuts just until blended (you may have to do this by hand with a wooden spoon.)
Drop the dough by heaping rounded tablespoonfuls onto the prepared baking sheet(s), spacing about 2-inches apart. Flatten the mounds slightly with the palm of your hand.
Bake until cookies are lightly golden around the edges, 10 to 12 minutes. Let the cookies cool on baking sheet a few minutes before transferring to a wire rack to cool completely.
Make the icing. In a medium bowl and using an electric mixer, beat the cream cheese, butter, and vanilla on medium speed until light and fluffy, 2 to 3 minutes, scraping down sides of bowl as needed. With the mixer on low speed, beat in the confectioners' sugar until creamy. Spread the icing over the cooled cookies and garnish with finely chopped nuts, if desired.