Step 1

Preheat the oven to 350°F. Butter a 9-inch-round cake pan. Line it with parchment and butter the parchment. To prepare the cake, whisk together the flour, baking powder, cinnamon, nutmeg, ginger, cloves, and salt in a medium bowl. In a large bowl, beat the 1/2 cup of butter and the 2/3 cup of brown sugar with an electric mixer on medium speed until pale and fluffy, 3 to 4 minutes. Beat in eggs and yolk, one at a time, until combined, and then stir in the vanilla. With the mixer on low, alternate adding the flour mixture and buttermilk to the butter mixture, starting with the flour. Set the batter aside while you finish the topping.

Step 2

To make the topping, melt the 1/4 cup butter in a skillet over medium heat. Add the 1/4 cup brown sugar, stirring constantly, until the mixture bubbles and starts to take on the texture of caramel, about 2 minutes. Remove from the heat and transfer the mixture to the prepared pan.

Step 3

Peel the carrots. Use the vegetable peeler to peel long, wide strips of carrot. Lay the strips to cover the bottom of the pan, overlapping them slightly. Finely chop any remaining strips of carrot — as well as any ends — and fold them into the cake batter. Top the carrot layer with the batter and smooth the top. Bake, rotating the pan halfway through, until golden-brown and a toothpick inserted in the center comes out with moist crumbs attached, 40 to 45 minutes. Let the cake cool in the pan on a rack for 5 minutes. Run a knife around the edge of the skillet and carefully invert the cake onto a serving plate. Serve warm or at room temperature.