Preheat oven to 350°F and line 2 baking sheets with parchment paper.
In a medium bowl, combine flour, oats, cinnamon, ginger, baking soda, and salt. Set aside.
In a large bowl or the bowl of a stand mixer fitted with a paddle attachment, cream together the butter, granulated sugar, and brown sugar on medium speed until fluffy. Mix in the carrots, then the egg and vanilla.
Add the dry ingredients and mix until fully incorporated, scraping the sides and bottom of the bowl as needed.
Scoop dough into balls and place on baking sheets, leaving space between to allow for rising.
Bake for 10-12 minutes, until golden. Let cool on the pans for 2 minutes, then transfer to a wire rack to cool completely.
To make the frosting
Beat together the butter and cream cheese until blended and smooth. Add the powdered sugar and beat until mixed, then beat in the vanilla. Taste and add more sugar if desired, or add milk 1 tablespoon at a time for a thinner frosting.
To assemble
Once cookies are completely cooled, spread frosting on the bottom of a cookie and top with another cookie of equal size to make a sandwich.
Store sandwiched cookies in the refrigerator. Unfrosted cookies can be stored at room temperature.
Notes
Nutrition information is for 1 cookie sandwich (2 total cookies plus frosting).