Put the apple juice with 3 tbsp lemon juice and vanilla sugar in a flat bowl and mix. Freeze the juice mixture for 3-5 hours, stirring every 30 minutes.
Wash mint, shake dry. Wash the apple, remove the core with an apple corer. Cut apple into thin slices and sprinkle with the rest of lemon juice.
Scrape the apple granite with a spoon from the bowl. Divide the apple slices into four martini glasses (approx. 200 ml each) and fill each with 2 cl Averna and 2 cl rum. Garnish with mint.