Preheat the oven to 350 degrees F. Butter and flour 2 (8-inch) round cake pans and line the bottoms with parchment paper. Butter the parchment well too.
Pulse the pineapple in a food processor fitted with a metal blade until it is finely chopped. Put the eggs in a bowl of a stand mixer fitted with the paddle attachment. Beat the eggs for 10 seconds. Add the oil, 1/4 cup melted butter, vanilla, carrots, chopped pineapple and grated fresh ginger and mix until combined.
Whisk together the flour, sugars, baking soda, cinnamon, ginger, nutmeg and salt in a bowl until combined. Add the wet ingredients and beat on low speed just until combined. Remove and fold in the crystallized ginger and pecans. Divide the batter between the 2 pans.
Bake until the tops are a very golden brown and a tester inserted in the center comes out with a few moist crumbs, 40 to 55 minutes. Let the cakes cool in the pans for 15 minutes, then invert them onto a wire rack and let them cool completely. Remove the parchment from the bottoms of each cake. Cool completely before frosting.
In a small saucepan fitted with a candy thermometer, bring the water, corn syrup, 1/3 cup plus 1 tablespoon sugar and vanilla bean to 246 degrees F. Before adding the syrup to the egg whites below, carefully remove the vanilla bean.
In the meantime, in a completely clean, dry mixing bowl of an electric mixer fitted with the whisk attachment, whip the egg whites, salt and cream of tartar until creamy and foamy, about 2 minutes. Still whisking, sprinkle in the remaining 1 tablespoon of sugar and continue to whisk until the whites hold very soft peaks, about 2 minutes. While mixing on slow speed, carefully drizzle in the hot syrup. Turn the mixer to high and whisk until thick, fluffy and just warm, about 7 minutes. Turn the mixer to low and whisk in the reserved vanilla seeds. Scrape the fluff into a bowl. Wash the bowl and whisk and return them to the machine.
Combine the cream cheese and butter in the bowl of the mixer and mix until light and fluffy. Add the powdered sugar and mix on low until combined. Increase the speed and whip until light and fluffy. Add about 2 cups of Fluff and the vanilla and whip until combined. Refrigerate for 15 minutes if too soft to use as frosting.