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

In a large bowl, cream 1 cup butter, brown sugar, and granulated sugar until light and fluffy, approximately 3 minutes.

Beat in the egg and 1 teaspoon vanilla extract until well combined.

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

Gradually add the dry ingredients to the wet ingredients, mixing until just combined.

Fold in the grated carrots and chopped walnuts using a spatula.

Drop rounded tablespoons of dough onto the prepared baking sheets, spaced 2 inches apart.

Bake for 10 to 12 minutes or until the edges are lightly golden and the centers are set.

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

For the frosting, beat cream cheese and 0.25 cup butter until smooth, then gradually add powdered sugar and 0.5 teaspoon vanilla.

Drizzle or spread the frosting onto the cooled cookies.