For the cake: Preheat the oven to 350 degrees F. Line a 9-by-2-inch round cake pan with parchment, and lightly coat with cooking spray. Toast the pecans on a small baking sheet until lightly browned and fragrant, about 8 minutes. Let cool, then coarsely chop.
Whisk together the flour, baking powder, cinnamon, salt and nutmeg in a medium bowl. Make a big well in the center so you can see the bottom of the bowl. Working within the well, add the sugar and egg and beat with a fork until well combined, then add the oil and vanilla and beat with the fork. Stir the dry and wet ingredients together with the fork until just combined. Fold in the carrots and pecans.
Transfer the batter to the prepared pan, and bake until a toothpick comes out clean, 20 to 25 minutes. Let the cake cool in the pan on a cooling rack for 20 minutes, then turn it out of the pan, remove the parchment and let cool completely, right-side up, on the rack, about 1 hour.
For the frosting: Process the cream cheese and butter in a food processor until completely combined and smooth, stopping once or twice to scrape the bottom and side of the bowl. Add the confectioners' sugar and vanilla, and process until just combined. Transfer to a small bowl.
To assemble: Cut the cake into 4 even wedges, as if you were cutting up a pie. Frost the top of each wedge with about 3 tablespoons of frosting (an offset spatula helps spread the frosting evenly). Stack the frosted wedges directly on top of each other on a serving plate, resulting in a 4-layer slice of cake with frosting between the layers and on top. Completely cover the curved back side of the cake with the remaining frosting, and leave the 2 remaining sides open. Refrigerate for at least 30 minutes. Cut the wedge in half, and serve.
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