Position racks in the top and bottom thirds of the oven; preheat to 350°F for light pans and 325°F for dark pans. Grease and flour three 8-inch cake pans.
For the cake, in a large bowl, whisk the flour, 2 cups sugar, the baking powder, salt and baking soda. In another medium bowl, whisk the buttermilk, butter, yolks, oil and vanilla. (If you really like coconut flavor, use 1 teaspoon coconut extract and 1 teaspoon vanilla extract).
In a medium bowl, using an electric mixer on high, beat the egg whites until very soft peaks form, about 2 minutes. Gradually beat in the remaining 1/2 cup sugar and beat until stiff, glossy peaks form.
Using an electric mixer on medium, gradually beat the buttermilk mixture into the flour mixture until just blended. Using a rubber spatula, fold half the egg-white mixture into the batter until almost blended. Fold in the remaining egg-white mixture and the sprinkles. Divide the batter among the three pans.
Bake the cakes, switching the pans between the top and bottom racks halfway through, until a toothpick inserted into the centers comes out clean, about 30 minutes. Let cake cool in the pans on a wire rack, about 15 minutes. If needed, run a thin knife around the edges of the pans to release the cakes, then invert them onto the rack and cool completely.
For the frosting, in a large bowl using an electric mixer on medium, beat the butter until smooth. Add the sugar, 1 cup at a time, blending to incorporate between additions. Beat in 1/4 cup milk and the vanilla. Increase the speed to medium-high; beat, adding milk by the tablespoon if too thick, until smooth and fluffy, about 5 minutes.
To assemble the cake, cut one of the three cakes in half.
From the center of the second cake, cut a 1- to 2-inch-wide strip. Cut two 1- to 2-inch-square pieces out of the strip. (You’ll use these pieces to make the tail.)
Using a 4-inch cookie cutter or a clean empty can, cut out two circles from the third cake. Cut off one-third of one of these small cake circles.
To create the rabbit’s body, stick the cake halves from the first cake together with frosting and place them cut-side down on the middle of a platter lined with wax paper. Using frosting, stick the smaller half-moon shapes leftover from the second cake on either side of the cake on the platter.
Using more frosting, stick two of the cake squares from the second cake side by side to one end of the cake to form the tail. Using frosting, stick the small cake circle from the third cake to the opposite end of the cake. Stick the cut circle to the full circle to form the head.
Frost the cake completely, building up the frosting where needed to even out the shape of the rabbit.
Sprinkle the cake with flaked coconut, pressing gently to adhere. (Want a brown bunny? Toast and cool the coconut first.) Carefully remove the wax paper.
Using frosting as glue, add Junior