Preheat oven to 350°F. Prep 3 9-inch round cake pans by lining them with parchment paper and spraying them with baking spray. Set aside.
In a medium-sized bowl, whisk together the flour, baking powder, and salt. In a large bowl, whisk together the sugar, vegetable oil, mango puree, milk, eggs, egg yolks, and vanilla. Whisk the flour mixture into the wet mixture just until the flour is incorporated. Do not overmix.
3⅔ cups all-purpose flour, 1 tablespoon baking powder, 1 teaspoon kosher salt, 2 cups granulated sugar, 1 cup vegetable oil, 1 cup mango puree, ¾ cup milk, 3 large eggs, 2 egg yolks, 2 teaspoons pure vanilla extract
Divide the cake batter into the prepared cake pans and place into the preheated oven to bake for 20-22 minutes or until a toothpick inserted comes out clean or with moist crumbs. Remove from the oven and place onto a cooling rack to cool to room temperature before frosting.
While the cakes are baking, make the compote. Combine all of the compote ingredients in a medium-sized saucepan set over medium heat. Cook, stirring constantly, for 6-8 minutes or until most of the liquid has cooked off. Remove from heat and allow to cool.
3 cups chopped fresh mango, ½ cup granulated sugar, 3 tablespoons lemon juice, ½ teaspoon kosher salt
While the cakes and compote are cooling, make the buttercream. Combine all of the buttercream ingredients in the bowl of a stand mixer fitted with a paddle or whisk attachment. Beat on low speed for a minute to allow the powdered sugar to incorporate. Increase speed to high and allow it to beat for 4-5 minutes or until a soft and smooth buttercream has formed.
1½ cups unsalted butter, 7½ cups powdered sugar, ½ cup mango puree, 1 teaspoon kosher salt
Fill a piping bag fitted with a round tip with ½ cup of buttercream. Fill a separate piping bag fitted with a star tip with buttercream.
Once the cakes and compote are sufficiently cooled, assemble the cake. Place the first layer of cake onto a cake stand or plate. Using the piping bag with the round tip, pipe a line around the edge of the cake to form a barrier for the compote. Pipe the ring two layers thick. Divide the compote and spoon half of it into the center of the frosting ring.
Place the second layer of cake on top of the frosting and compote. Pipe another frosting ring on top of the second layer and fill with the remaining compote. Place the third layer on top. Frost the top and sides of the cake with a thin layer of buttercream.
Starting in the center of the cake, using the piping bag fitted with the star tip, pipe a blob about the size of a quarter. Use the back of the spoon and press into the center of the blob and pull outwards in a swirl. Pipe another blob close to the remainder of the previous blob. Using the spoon, press into the center of the blob and pull outward. Continue this pattern in a swirl shape around the top of the cake until you reach the edge. Once you reach the sides, simply pipe and pull in a line around the cake. You should be able to have about 3 layers of blobs on the sides.
Once the cake is frosted, slice and serve!