Special equipment: a pastry bag fitted with a round tip, optional
For the cake: Preheat the oven to 350 degrees F. Line three 9-inch round baking pans with parchment and spray the sides with cooking spray.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon and ginger.
In a separate, large bowl or in the bowl of a stand mixer, beat together the brown sugar and oil until fully combined, 2 to 3 minutes. Add the eggs one at a time and then the vanilla, scraping down the sides if necessary to ensure that all is thoroughly combined. Gently stir in the flour mixture until just combined. Stir in the carrots, pineapple and pecans until they are evenly distributed, taking care not to overmix the batter.
Pour a third of the batter into each of the cake pans and bake until the cake is fully set and a toothpick inserted in the center comes out clean, 40 to 50 minutes. Cool completely in the pans on a rack, about 1 hour.
For the frosting: Meanwhile, beat the cream cheese, butter, powdered sugar, orange extract and orange zest with an electric mixer in a medium bowl until smooth and fluffy, 2 to 3 minutes.
When completely cool, remove the cake layers from the pans. Place one layer on a cake plate. Using an offset spatula, spread a quarter to a third of the frosting over the top of the layer, taking care to spread it all the way to the edges. Top with a cake layer and repeat with the remaining frosting and cake layer, smoothing the frosting neatly over the top layer. Scatter zest over the frosting on the top layer.
If a more uniform finish is desired, use a pastry bag fitted with a round tip to pipe additional frosting into the remaining gaps between the layers, turning the plate as you go. Use the end of a spoon or the tip of a knife to smooth the rounded edge of the piped frosting and close any gaps.
To protect the exposed rim of the cake plate from crumbs and frosting smudges as you work, begin by cutting wide strips of parchment paper and placing them under the edges of the bottom cake layer, overlapping and covering the exposed rim of the cake plate. When the cake is fully assembled and ready to serve, carefully slide out the parchment strips to expose the clean rim of the plate.