Start by preheating your oven to 350°F. Line a baking sheet with parchment paper or lightly butter it. This prevents cookies from sticking and ensures even baking.
Beat butter with brown and granulated sugars until fluffy, about three minutes. Add eggs one at a time, mixing just until combined, then stir in vanilla. This creates a smooth, rich base for your dough.
Pulse ¾ cup of oats into coarse oat flour. In a large bowl, whisk oat flour, remaining oats, flour, baking soda, and spices together. This ensures even flavor and texture in each bite.
Stir the dry ingredients into the wet mixture with a wooden spoon. Fold in the shredded carrots until evenly distributed. Don’t overmix, or cookies may turn tough.
Drop dough by rounded tablespoonfuls onto the cookie sheet. Leave them rounded and slightly messy for the best texture. Bake for 10 minutes, then let cool on the sheet for a few minutes before transferring to a wire rack.
Beat softened butter and cream cheese until smooth. Add vanilla, then powdered sugar gradually, followed by heavy cream until creamy. Spread frosting on half of the cookies and sandwich with remaining cookies.