Chop dark chocolate. In a heatproof bowl, melt some of the chocolate over a saucepan filled with simmering water. Meanwhile, inflate and knot the balloons. Remove chocolate from heat and stir in the remaining chocolate until melted and smooth.
Grease the balloon with a bit of oil. Dip balloon into the chocolate and gently turn to coat. Place each balloon on a cup, chocolate covered-side up, and refrigerate for at least 30 min.
For the carrot cakes, pre-heat oven to 180°C/360°F. Separate egg white from egg yolk and transfer to a stand mixer. Add salt and ground cardamom. Scrape out vanilla bean seeds and add to the egg whites, together with the sugar. Beat until stiff. Reserve the vanilla bean pod. Roughly grate carrots and zest lemon. Transfer grated carrots and lemon zest to a bowl, then juice lemon into the bowl. Add ground almonds and flour to a bowl and stir to combine. Add egg yolks to the beaten egg white and stir in ground almond-flour mixture and grated carrots.
Grease ramekins with butter, then dust with flour before dividing batter between the ramekins. Bake at 180°C/360°F for approx. 20 – 25 min.
Peel, quarter, and core apples, then cut into pieces. Transfer to a small saucepan and gently cook over medium heat for approx. 20 – 25 min., or until soft.
Add almond milk and reserved vanilla bean pod to a saucepan and bring to a simmer. Chop white chocolate and stir into the almond milk until melted. Keep warm until served.
Remove carrot cakes from the oven and allow them to cool for approx. 10 min. Meanwhile, crush soft cooked apples with a fork. Then unmold carrot cakes, brush with apricot jam, and garnish with slivered almonds.
Once the chocolate has dried, gently loosen the edges from the balloon. Deflate balloon with a knife, then remove it. Dust chocolate shell with edible gold dust. Remove vanilla bean from the almond-chocolate sauce and transfer it to a sauce boat. Serve carrot cakes on a plate and top with stewed apples. Place the chocolate eggshells over each cake. Pour warm white chocolate-almond sauce over it and enjoy!