Preheat oven to 350 degrees F. Grease an 8x8 square cake pan and line with parchment paper.
Wash, peel, and grate carrots. It took me about 4 large carrots to get 1 1/2 cups grated.
In a medium bowl, whisk together flour, baking soda, ground cinnamon, ground ginger, fresh grated nutmeg, and salt. Set aside.
Stir together 1/3 cup of 0% plain Greek yogurt with water until yogurt reaches the consistency of a thick cream. You should pour freely. Set aside.
In bowl of a stand mixer with paddle attachment, combine oil, light brown sugar, dark brown sugar, and vanilla extract. Turn stand mixer on to medium and beat until combined. Add eggs one at a time. Alternating between the Greek yogurt and flour, add wet and dry ingredients to the sugar and egg mixture until just combined. Pause stand mixer periodically to scrape down edges.
Using a spatula, fold in grated carrots and chopped walnuts. Pour batter into cake pan. Bake in middle rack for 35-40 minutes or until toothpick inserted into center of cake comes out clean.
Let cake cool on a wire rack completely before frosting. You can also store unfrosted cake in the freezer.