Preheat the oven to 350F/175C and line a large baking sheet with a silicone mat or parchment paper. Set aside.
Add 1/2 cup (1 stick) room temperature butter, 1/4 cup room temperature cream cheese, 2/3 cup packed light brown sugar, and 1/3 cup granulated sugar into a large bowl. Beat on a medium speed for about 30 seconds until the mixture is smooth and no lumps remain (we don't need to cream the mixture as much as we do when making cakes). If you notice any stubborn brown sugar chunks, use the back of a fork to break them up.
Add 1 large egg and 1 tsp vanilla extract or paste. Mix on a medium speed until fully combined. If you want the dough to have a hint of an orange color, you can add a small drop of orange gel food coloring in this step.
Sift 1 2/3 cups flour (make sure it's properly measured by using a scale or fluffing the flour in the bag, then spooning it into your measuring cup and leveling with a knife), 1/2 tsp fine salt, 1/4 tsp baking soda, 1 1/2 tsp cinnamon, 3/4 tsp ginger, 1/2 tsp ground nutmeg, 1/8 tsp cloves into the butter mixture and mix on a low speed until no visible streaks of flour remain. Scrape the sides and bottom of the bowl with a rubber spatula (hidden pockets of butter will cause the cookies to overspread!). If you don't have a sifter, whisk the dry ingredients together in a separate bowl, then pour them into the wet ingredients.
Place 1/2 cup of finely grated carrots on a kitchen towel or a few pieces of paper towel and squeeze out any excess liquid (my carrots started at 70g, and were 48g after I wrung out them out). Save 1 Tbsp of the carrots to garnish the cookies with, and fold the remaining carrots into the dough. Scrape the sides and bottom of the bowl as needed with a rubber spatula.
Use a measuring spoon or cookie scoop to shape 1 1/2 Tbsp cookie dough balls. You should end up with about 22 cookie dough balls. Then press a small pinch of the reserved shredded carrots on top of each cookie.
Bake the cookies one tray at a time for 12-14 minutes (the sweet spot in my oven is 13 minutes, but every oven bakes differently), and rotate the pan halfway through to help them bake evenly. The cookies are ready when the edges just start to brown, but the centers still look puffy and slightly under-baked. The cookies will continue to bake on the sheet and firm up once removed from the oven, so it's always best to err on the side of under-baked!
Place the cookie sheet on a wire rack and let the cookies cool fully on the pan. Let the cookies cool fully before adding the cream cheese drizzle.