Set out butter and cream cheese for the frosting and let both come to room temperature.
Line three baking sheets with parchment paper and set aside.
Mix dry ingredients and set aside. Peel and grate carrots and set aside.
In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl (if using a hand mixer), cream butter and both sugars. When smooth, add egg, vanilla, and grated carrots and mix until combined.
Add flour mixture to the wet ingredients and mix until just combined.
Using a large, 2-ounce cookie scoop, place 6 level scoops of cookie dough on each prepared baking sheet, evenly spaced.
Chill cookie dough on cookie sheets for 30 minutes in the refrigerator. **see note
Part of the way through chill time, preheat oven to 350 degrees.
Place prepared and chilled cookie sheets in the preheated oven and bake for 28 minutes. While the cookies bake is a perfect time to wash your mixing bowl and attachment to reuse for the frosting.
Halfway through bake time, pull trays out to flatten the cookies with the bottom of a glass, then rotate and swap racks when you place back in the oven to finish baking.
Cookies are finished baking when the edges of the cookies turn a nice, golden brown.
Transfer each cookie sheet to a wire rack and let the cookies cool completely on the tray. While the cookies cool, make the frosting.
Sift the powdered sugar using a fine mesh strainer and set aside. This will remove lumps from your powdered sugar.
Using a stand mixer fitted with the paddle attachment or a handheld mixer and a large mixing bowl, cream together the remaining 1 stick of butter and cream cheese until completely smooth. Add in powdered sugar, a little bit at a time to avoid a mess, and mix at a low speed after each addition. Scrape down the sides of the bowl as needed. Add in salt and vanilla extract and mix at a low speed until combined. Increase to medium-high speed and mix for 2-3 minutes until frosting starts to lighten in color.
Use a medium cookie scoop to place one scoop of frosting on each of the cooled cookies and smooth out using an offset spatula.