Preheat the oven to 350°F and prepare two 8-inch round cake pans by greasing, flouring, and lining them with parchment paper.
Finely chop the pineapple using a food processor and set it aside.
In a stand mixer, briefly beat the eggs, then add the oil, melted butter, vanilla, carrots, chopped pineapple, and grated fresh ginger. Mix until well combined.
In a separate bowl, whisk together the flour, sugars, baking soda, cinnamon, ginger, nutmeg, and salt.
Add the wet mixture to the dry ingredients and mix until just combined.
Gently fold in the crystallized ginger and pecans.
Divide the batter equally between the prepared pans and bake until golden brown and a toothpick inserted into the center comes out clean, about 40 to 55 minutes. Let the cakes cool in the pans for 15 minutes before transferring them to a wire rack to cool completely. Remove the parchment paper before frosting.