Preheat the oven to 400°F. In a bowl, mix the flour, baking powder, baking soda, and salt. Mix the butter, oil, sugar, brown sugar, and lemon juice in a separate bowl until creamy, then whisk in the eggs one by one. Now mix in the dry ingredients with the milk and stir until it turns into a homogeneous batter. Transfer the batter to a greased baking pan lined with parchment paper and bake for 35 minutes until golden brown, then let the cake cool down completely and remove it from the baking pan.
Cut the dome off the cake. Cut off the edge of the dome. It's best to use a bowl to get the round shape of the dome.
Use a straw to make a circle of holes in the cake.
Hollow out the inside of the cake. The hollowed-out area should have the same diameter as the cut-out cake dome, so use the same bowl from earlier to do so.
Mix the part of the cake you removed with the butter and spread it over the bottom half of the cake, then refrigerate the cake for 30 minutes to allow the coating to set.
Meanwhile, start on the blue glaze. Bring the water and sugar to a boil, ideally bringing the mixture to 215°F to give the glaze a beautiful shine. Once you reach this temperature, immediately remove the pan from the heat and stir in the white chocolate with a whisk. Add the syrup and heavy cream – the mixture should have a creamy consistency by now. As soon as the temperature of the glaze drops to 140°F, gently mix the gelatin and dissolve it in the glaze. Puree the mass, making sure not to work in any air to avoid getting bubbles. As you're pureeing the glaze, add blue food coloring. The glaze should be about 86°F before putting it on the cake. If you don't feel like getting the thermometer out, you can also just mix the glaze as shown in the video, but be prepared for the fact that it won't look quite as shiny this way. Place the cake on an elevated surface, like a deep plate, and fix the cake in place with the edge of a springform pan.
Pour the glaze through the sieve into the cake. The hollowed-out part should be filled about 3/4 full with the glaze.
Place the perforated cake dome carefully on top of the glaze and then press it down slowly and evenly so that the glaze starts dripping out of the holes. Except for a small island in the middle, the entire springform pan should be filled with blue glaze.
Wait until the glaze has set and then carefully remove the springform pan. Decorate the cake with chopped pistachios, pocky sticks, and mint leaves to create an inviting island!