3 tablespoons unsalted butter, softened, plus more for cookie sheets
3/4 cup all-purpose flour
1/2 teaspoon baking soda
1/4 teaspoon kosher salt
1/2 teaspoon cinnamon
1 teaspoon ground ginger
Pinch ground cloves
1/8 teaspoon freshly, finely grated nutmeg
1/8 teaspoon freshly, finely ground black pepper
6 tablespoons vegetable oil
1/2 cup firmly packed light brown sugar
1 large egg
1/4 cup sour cream
3/4 teaspoon vanilla, divided
1 1/2 medium carrots, peeled and shredded on the fine side of a box grater (to yield 1 cup)
1/4 cup unsweetened shredded dried coconut
5 ounces cream cheese, softened
2/3 cup powdered sugar
1 teaspoon heavy cream or milk
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.