FOR THE CAKE:
Olive oil for the pans
1 cup sugar
1 cup firmly packed light brown sugar
3/4 cup olive oil
9 oz. (2 cups) all-purpose flour, sifted
2 tsp. ground cinnamon
1 tsp. freshly grated nutmeg
2 Tbs. baking powder
3 cups grated carrots (or pulse in food processor until finely chopped)
1/2 tsp. salt
4 large eggs, at room temperature
2 tsp. vanilla extract
1 cup toasted pecans, finely chopped
1/4 cup dark rum
FOR THE FROSTING:
1 Tbs. grated orange zest
1/2 cup honey
two 8-oz. packages cream cheese, softened
1/2 cup heavy cream
To make the cake — Position a rack in the middle of the oven. Heat the oven to 350°F. Oil two 9×2-inch cake pans with olive oil, line the bottoms with parchment or waxed paper, and oil the paper.
Put both sugars and the olive oil in the bowl of a stand mixer and set aside. In a medium bowl, combine the flour, cinnamon, nutmeg, baking powder, and salt, mix well, and set aside. Grate the carrots, measure 3 cups of carrots and set aside. In a small bowl, lightly beat the eggs with a fork, stir in the vanilla, and set aside.
Beat the sugar mixture on low until well combined, (it will look like wet sand). Continuing on low speed, gradually mix in half the dry ingredients. Add the remaining dry ingredients in 3 or 4 additions, alternating with the egg mixture, and ending with the dry; scrape the sides of the bowl once or twice. Stir in the carrots, pecans, and rum. Let the batter sit for 15 min.
Divide the batter between the cake pans and bake until a toothpick inserted in the center of each comes out clean, 35 to 40 min. Cool them in the pans on a rack for 15 min. Run a paring knife around the inside edge to release the cakes. Carefully turn each pan over so the bottom faces up, remove the pan, and carefully peel off and discard the paper liner. Flip each layer over so the top faces up again. Let cool completely.
To make the frosting — When the layers are cool, put the cream cheese, honey, and orange zest in the bowl of a stand mixer and whip on high until smooth and light, 1 to 2 min. Add the cream and whip on medium, scraping the sides of the bowl, just until you see tracks from the whip or beaters, 1 to 2 min.