Preheat the oven: Set the oven to 350°F (175°C). Line the bottoms of three 6-inch cake pans with parchment paper and set aside.
Mix dry ingredients: In a large bowl, sift together the cake flour and baking powder. Whisk to combine, then set aside.
Make the meringue: In a separate large bowl, use a stand mixer or hand mixer to beat the egg whites until foamy. Add cream of tartar and beat on high until frothy. Gradually add ⅓ of the sugar (¼ cup), one tablespoon at a time, until stiff peaks form.
Prepare the egg yolk mixture: Beat the egg yolks, remaining sugar (½ cup), salt, milk, and vanilla extract on high speed until pale yellow and fluffy.
Combine dry ingredients and egg yolk mixture: Fold the dry ingredients into the egg yolk mixture in three additions, being careful to mix gently.
Incorporate oil: Mix a few spoonfuls of the egg yolk mixture with the vegetable oil, then fold it back into the remaining egg yolk mixture.
Fold in meringue: Gently fold the egg white meringue into the egg yolk mixture in three additions, ensuring there are no streaks.
Bake: Divide the batter evenly between the prepared cake pans. Tap the pans gently to release air bubbles. Bake for 20-22 minutes or until a toothpick comes out clean. Let the cakes cool for 10 minutes before releasing them from the pans, then cool completely on a wire rack.
Whip the cream: In a large bowl, beat the whipping cream, powdered sugar, and vanilla until medium-stiff peaks form. Chill until ready to assemble.
Assemble the cake: Place the first layer of cake on a turntable or flat surface. Spread a layer of whipped cream on top, followed by a layer of mango chunks. Add another layer of whipped cream on top of the mangoes. Repeat with the remaining layers.
Decorate: Apply a thin layer of whipped cream as a crumb coat, and chill the cake for 15 minutes. Apply the final layer of whipped cream and decorate with a piping bag and additional mango chunks.
Chill before serving: Refrigerate the cake for 1-2 hours before slicing and serving.