Combine the sugar, juice of 1 lime and 1/2 liter (approximately 2 cups) of water in a saucepan and boil until syrupy.
For garnish, rinse 2 apples, halve, remove seeds and cut into thin slices. Poach in half of the syrup for 2-3 minutes. Let cool in the syrup. Remove the apple slices gently, lay on a baking sheet lined with parchment paper and allow to dry at 50°C (approximately 120°F) in a preheated oven. Then let cool.
Rinse remaining apples, remove stems and set stems aside. Cut apples into quarters, remove the seeds and puree with the remaining lime juice. Stir in half of syrup. Freeze in an ice cream machine.
Place a scoop of apple sorbet in the middle of each plate and garnish with the apple slices in a fan shape. Mix remaining sorbet with the Calvados and pour over the plates. Garnish with the apple stems and serve.