Preheat oven to 350°F (180°C). Spray a 17½×12½-inch rimmed baking sheet with baking spray with flour. Line bottom of pan with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, beat sugar and neutral oil at medium speed until moist and well combined, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition. Beat in vanilla.

In a medium bowl, whisk together flour, baking powder, and salt. With mixer on low speed, gradually add flour mixture to sugar mixture alternately with milk, beginning and ending with flour mixture, beating just until combined after each addition. Spread batter in prepared pan.

Bake until top is dry, edges are lightly golden, and a wooden pick inserted in center comes out clean, 15 to 20 minutes. Let cool completely in pan.

Freeze cooled cake until firm, about 30 minutes.

Using a 2½-inch round cutter, cut 28 rounds out of cake, discarding scraps.

Place White Chocolate Swiss Meringue Frosting in a pastry bag fitted with a large round piping tip (Ateco #805). Pipe frosting on top of half of rounds. Place remaining rounds on top of frosting.

In a small microwave-safe bowl, heat white chocolate on high in 10-second intervals, stirring between each, until melted and smooth (1 to 1½ minutes total). Stir in coconut oil until well combined.

Line 2 baking sheets with parchment paper. Place a wire rack on 1 prepared pan. Place cakes on prepared rack. Slowly pour white chocolate mixture over cakes until evenly coated. Gently tap pan with rack on counter so excess white chocolate mixture drips off. Using a small offset spatula, transfer coated cakes to remaining prepared pan. Refrigerate until firm, about 20 minutes.

Drizzle cakes with melted dark chocolate. Refrigerate until firm, about 10 minutes. Refrigerate in an airtight container for up to 1 week.

In the heatproof bowl of a stand mixer, combine sugar, egg whites, and salt. Place bowl over a saucepan of simmering water, and cook, whisking frequently, until sugar dissolves and an instant-read thermometer registers 130°F (54°C).

Carefully return bowl to stand mixer. Using the whisk attachment, beat at high speed until fluffy and bowl is cool to the touch, 6 to 8 minutes. Add butter, 1 tablespoon (14 grams) at a time, beating until combined after each addition. (If buttercream breaks, beat for 2 to 3 minutes, and emulsion will come back together.) (See Kitchen Tips.) Reduce mixer speed to medium, and beat until smooth and shiny. Add melted white chocolate, and beat just until combined.