To make the cake -- Position a rack in the middle of the oven. Heat the oven to 350°F. Oil two 9x2-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. In a food processor fitted with the metal blade, process the carrots until they're in tiny pieces, scraping down the sides of the bowl, about 25 seconds. 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.
Mild and inexpensive, plain olive oil -- not extra-virgin -- is the wiser choice for cakes.
Beat the sugar mixture on low until well combined, scraping down the sides of the bowl once, 2 to 3 minutes (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, scraping the sides of the bowl once. Let the batter sit for 15 minutes.
The last additions -- carrots, pecans, and rum. Let the batter rest so the flavors can permeate it.
Divide the batter between the cake pans (if you have a scale, weigh them to see if they're even) and bake until a toothpick inserted in the center of each comes out clean, 35 to 40 minutes. Cool them in the pans on a rack for 15 minutes. Run a paring knife around the inside edge to release the cakes. With the help of a second rack, turn each pan over so the bottom faces up, remove the pan, and carefully peel off and discard the paper liner. Using the racks again, 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 minutes, scraping the sides of the bowl. 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 minutes.
To frost the cake -- Set one cake layer on a cardboard base or other support (like a removable tart pan bottom) and spread it evenly with about one-third of the frosting. Set the second layer on top and cover the top smoothly (or with little swirls) with about one-third more of the frosting. Coat the sides evenly with a very thin layer of frosting, and then use what remains to finish the sides with a second coat. Refrigerate the cake for several hours-this firms up the frosting and mellows the flavors-but give it some time at room temperature before serving to take off the chill.