Line a sheet pan with parchment paper or a silpat liner. Don't skip this step! Set aside until ready to use.
Peel and then finely grate the carrots. Measure lightly packed into 1 cup measuring cup and then give the finely grated carrots a quick coarse chop and set aside.
In a large bowl, combine the room temperature (NOT melted) butter, brown sugar and white sugar. Beat at medium speed, scraping bowl often, until smooth and creamy (should look like peanut butter). Add in the egg and vanilla and beat until combined.
In another bowl, whisk together the flour, baking soda, cinnamon, nutmeg, ginger and salt. Add to the wet ingredients and beat together until JUST combined. Stir in the finely grated carrots. Cover the dough and refrigerated for 1 hour.
Preheat the oven to 350 degrees F. Roll out balls (the dough is still quite sticky and not as firm as regular cookie dough so it gets messy) that are a tablespoon and a half of dough. Don't put more than 8 cookies on a sheet pan so they have plenty of space. Bake 9-13 minutes or until lightly browned around edges (be careful to not overbake them. Slightly underbaked cookies will remain soft and cake-like. Cool completely.
Meanwhile prepare the frosting. In a large bowl, beat together the room temperature butter and cream cheese until completely smooth. Add in the vanilla, cinnamon, and salt. Beat to combine. Add in the powdered sugar and then beat until the frosting is smooth (it will seem like you need more liquid, but just keep beating, it will come together!)
Frost each cooled cookie with frosting and enjoy! Store any leftover cookies in an airtight container in the fridge.