Preheat oven to 350°. Put a dab of butter in each corner of 2 large baking sheets and line with parchment.
In a medium bowl, whisk together flour, baking soda, salt, and spices. In a large bowl, whisk together oil, brown sugar, egg, sour cream, and 1/4 tsp. vanilla; stir in dry ingredients with a wooden spoon until well combined. Add carrots and coconut, stirring until just combined.
Drop 1-tbsp. circles of batter onto baking sheets about 1 in. apart. Bake until cooked through and browned slightly, 14 to 18 minutes. Cool on baking sheets.
Using a stand mixer or hand mixer, beat cream cheese until very soft. Add butter and beat until smooth and well blended. Add powdered sugar a third at a time, beating until smooth after each addition. Beat in cream and remaining 1/2 tsp. vanilla.
Flip cakes over and spread about 1 1/2 tbsp. frosting on half of them. Top each with a remaining cake, flat side down, to sandwich the frosting.
Note: Nutritional analysis is per cake.