Preheat the oven to 170° traditional heat. Grease and line a 30x20cm rectangular mold like me or a 20x20 mold and a 10x20 mold (we will ultimately obtain a 10x20cm Neapolitan).
If you use a rectangular mold like me, divide it into two parts by adding folded baking paper (20x20 on one side and 10x20 on the other) to cook the two vanilla and chocolate doughs at the same time (see my photos step by step).
Separate the whites from the yolks.
Whisk together the egg yolks, powdered sugar and liquid cream for about 3 minutes, until the mixture turns white.
Then add the cooled melted butter and mix. Then add the milk and oil together and mix.
Separately, sift together the flour, baking powder and salt. Then add twice to the dough. Do not overmix: just enough so that you no longer see the flour in the dough.
Beat the egg whites with a mixer then add them to the dough in three batches using a spatula so as not to damage the egg whites.
Separate the dough by dividing 2/3 on one side and 1/3 on the other. In 2/3 of the dough, add the vanilla and mix gently. In 1/3 of the dough, incorporate the bitter cocoa, still using a spatula.
Pour the cocoa paste into the smallest part of the mold (1/3) and pour the vanilla paste into the largest part of the mold (2/3).
Bake for around 15-18 minutes: the sponge cakes should be plump and slightly golden. Then leave to cool completely in the mold before unmolding so as not to break the sponge cakes.
Chocolate ganache
Heat the liquid cream in a saucepan, as soon as it comes to the boil, pour in two batches over the chocolate cut into pieces. Mix to obtain a smooth and shiny ganache. Add the butter in small pieces and smooth. Cover the ganache with film and place in the refrigerator for approximately 30 minutes/1 hour: until it takes on the consistency of a spread.
Operating manual
Cut the tops of the cakes if they are not quite flat. Cut the vanilla sponge cake in half to obtain 3 strips of cakes (see step-by-step photos)
Place the first vanilla strip, cover it generously with chocolate ganache with a spatula. Place the chocolate strip and press lightly. Place ganache with a spatula. Place the last vanilla strip. Check that everything is straight. Wrap everything tightly in cling film and refrigerate for 1 hour.
Then trim the edges of the Neapolitan with a long serrated knife to obtain clean, smooth edges.
White icing
Mix the icing sugar and milk to obtain a fairly thick mixture that can still be spread. If necessary add a few drops of milk: do not add too much to avoid it being too liquid (if this is the case add icing sugar to balance).
With a tablespoon, place a thin layer of icing on top of the Neapolitan and spread to the edge, taking care not to drip any over the sides.
Sprinkle with chocolate sprinkles. Allow the white icing to set at room temperature before serving.
Notes
Neapolitan is a cake that keeps at room temperature for up to 3 days, in an airtight container. This prevents it from drying out and therefore retains all its softness.