Make cake: Preheat oven to 350°. Grease 3 round cake pans (8 in. each). Line each with a circle of parchment paper and set aside.
Mix cake flour, whole-wheat flour, baking soda, baking powder, salt, cinnamon, cardamom, and star anise in a large bowl; set aside.
In bowl of a stand mixer fitted with paddle attachment, beat oil, eggs, coconut sugar, yogurt, orange juice, bourbon, and vanilla on medium speed until well blended. Add flour mixture; beat on low speed to blend, then on medium speed until smooth, scraping inside of bowl twice. Stir in orange zest and crystallized ginger, then squash.
Divide batter among lined pans and spread level. Bake until a toothpick inserted in center comes out clean, 20 to 24 minutes.
Turn cakes onto racks, remove parchment, and let cool completely, 1 to 1 1/2 hours.
Make frosting: In bowl of a stand mixer fitted with paddle attachment, beat butter and cream cheese until very smooth. Add powdered sugar, bourbon, cardamom, salt, cinnamon, and ginger; beat on low speed to blend, then on high speed until smooth and fluffy, scraping inside of bowl once or twice.
Set a cake layer rounded side down on a platter; spread top of it with 3/4 cup frosting. Repeat with second layer. Set top layer in place, rounded side up. Spread top and sides of cake with remaining frosting. Press coconut into sides, a small handful at a time.
*Find coconut palm sugar in the baking aisle of well-stocked grocery stores. Toast coconut on a rimmed baking sheet in a 350° oven until light golden, stirring once, 4 to 5 minutes.
Make ahead: Up to 1 day, chilled and loosely wrapped with plastic wrap.