Instructions

for the cake

Adjust oven rack to lower-middle position and heat oven to 350 degrees. Grease two 9-inch cake pans, line with parchment paper, grease parchment, and flour pans. Whisk flour, 1 1/4 cups sugar, baking powder, baking soda, and salt together in large bowl. Whisk buttermilk, egg yolks, melted butter, oil, and vanilla together in separate bowl. Add buttermilk mixture to flour mixture and whisk until just combined.
Using stand mixer fitted with whisk attachment, whip egg whites and cream of tartar on medium-low speed until foamy, about 1 minute. Increase speed to medium-high and whip whites to soft, billowy mounds, about 1 minute. Gradually add remaining 1/4 cup sugar and whip until glossy, stiff peaks form, 2 to 3 minutes.
Whisk one-third of whites into batter. Using rubber spatula, gently fold remaining whites into batter until no white streaks remain. Divide batter evenly among prepared pans, smooth tops, and gently tap pans on counter to release air bubbles.
Bake until tops are light golden and toothpick inserted in center comes out clean, 20 to 25 minutes, rotating pans halfway through baking. Let cakes cool in pans for 15 minutes. Remove cakes from pans, discard parchment, and let cool completely on wire rack, about 2 hours.

for the filling and frosting

Grind tapioca to fine powder in spice grinder, about 30 seconds. Coarsely chop 2 pineapple quarters, then pulse in food processor until coarsely ground, 8 to 10 pulses (you should have about 1½ cups). Slice one of remaining pineapple quarters into 1/8-inch-thick slices and reserve to decorate cake. (Eat remaining pineapple quarter—cook’s treat!) Transfer ground pineapple to medium saucepan and add granulated sugar, rum, ground tapioca, and 1/8 teaspoon salt. Bring to boil over medium heat and cook until thickened, about 4 minutes. Off heat, stir in 2 tablespoons butter until incorporated. Transfer to bowl and refrigerate until set, at least 4 hours or up to 24 hours.
Using stand mixer fitted with whisk, whip remaining 18 tablespoons butter, coconut milk, vanilla, coconut extract, and remaining ⅛ teaspoon salt on medium-low speed until combined. Slowly add confectioners’ sugar and continue to whip until smooth, about 2 minutes. Increase speed to medium-high and whip until light and fluffy, about 5 minutes.
Place 1 cake round on plate or pedestal. Spread filling evenly over top. Top with second cake round, press lightly to adhere, then spread frosting evenly over top and sides of cake. Gently press coconut onto sides of cake. Fan pineapple slices in circle around top edge of cake. Serve.