Before making the cookie dough, shred the carrots and set them aside on a paper towel. The paper towel absorbs excess moisture from the carrots.
In a large mixing bowl, cream the butter, granulated sugar, and brown sugar together with an electric mixer for about 2 minutes. Mix in the eggs one at a time and then add the vanilla extract. Add in the shredded carrots and old-fashioned oats.
In a separate medium-sized bowl, sift together the flour, salt, and baking soda. Add the ground cinnamon and ginger to the bowl of dry ingredients. Gradually add the dry ingredients to the large bowl and mix until just combined. Use a rubber spatula to scrape the sides of the bowl.
Line a baking sheet with parchment paper. Scoop out 2-tablespoon-sized cookie dough balls onto the baking sheet and chill them in the fridge for 20-30 minutes. You should have 36 cookies total (for 18 cookie sandwiches). Preheat oven to 350℉.
Once chilled, bake the cookies on the parchment paper-lined baking sheet. Separate the dough balls by about 2 inches. Bake for 12-15 minutes or until the cookies are golden brown around the edges but a bit undone in the middle. *See note below for perfectly round cookies.* Let the cookies cool on a wire rack while you make the frosting for the filling.