Move an oven rack to the center position and preheat the oven to 350°F (175°C). Grease the bottom and sides of both 9-inch round cake pans with 1 tablespoon of butter, press a parchment paper round into each pan, then lightly grease the parchment.
Spread the 1 1/2 cups of topping coconut in a single layer on a rimmed baking sheet and bake for 5 to 8 minutes, stirring every 2 minutes, until golden brown and nutty. Watch closely at the end, then slide onto a cool plate to cool completely.
Peel the ripe bananas and mash them with a fork or potato masher in a medium bowl until mostly smooth with a few small lumps. Measure out 1 1/2 cups of mashed banana and set aside.
In a large bowl, whisk together the flour, baking powder, baking soda, and 1/2 teaspoon salt for about 30 seconds so the leavening is evenly distributed. Set aside.
In a separate large bowl, beat the 3/4 cup softened butter and granulated sugar with an electric mixer on medium speed for 3 to 4 minutes, until pale and fluffy. Scrape down the sides and bottom of the bowl halfway through.
Add the eggs one at a time, beating about 30 seconds after each until blended, then beat in the 1 1/2 teaspoons vanilla and 1 teaspoon coconut extract. Add the mashed bananas and mix on low just until combined; the batter may look slightly curdled, which is normal.
With the mixer on low, add one third of the flour mixture, then half the buttermilk, then another third of the flour, the remaining buttermilk, and the last of the flour. Stop mixing as soon as no dry streaks remain, because overmixing makes the cake tough.
Add the 1 cup of shredded coconut for the batter and fold it in gently with a rubber spatula until evenly distributed, scraping the bottom of the bowl once to catch any hidden flour.
Divide the batter evenly between the prepared pans, smooth the tops with the back of a spoon, and tap each pan gently on the counter to release large air bubbles. Bake for 28 to 32 minutes, until golden and a toothpick inserted in the center comes out clean or with a few moist crumbs; tent with foil for the last 8 minutes if browning too quickly.
Let the cakes cool in their pans on a wire rack for 15 minutes, run a thin knife around the edges, then flip each cake onto the rack and peel off the parchment. Cool completely, about 45 minutes, before frosting.
Beat the 8 ounces softened cream cheese and 1/2 cup softened butter on medium speed for 2 minutes until smooth. Reduce speed to low and add the sifted powdered sugar 1 cup at a time, then add 1 teaspoon vanilla, 1/2 teaspoon coconut extract, and the pinch of salt. Beat on medium-high for 1 to 2 minutes until fluffy, adding coconut milk 1 tablespoon at a time if too stiff to spread.
Place one cake layer on a serving plate or cake stand, flat side up, and spread about 1 cup of frosting over the top all the way to the edges. Place the second layer on top, flat side down, press lightly so it sits level, then spread a thin layer of frosting over the top and sides to seal in the crumbs and chill for 15 minutes.
Cover the cake with the remaining frosting, making soft swoops on top with an offset spatula, then press the cooled toasted coconut gently onto the sides and sprinkle it over the top. Chill for 20 minutes to set the frosting, then slice with a sharp knife wiped clean between cuts.