In a small bowl, combine the caster sugar and 2 tbsp cold water. Stir until the sugar begins to dissolve. Add the grated ginger and let sit for 5 minutes so the ginger releases flavor into the syrup.

Blend fruit

Place the raspberries and fresh lemon juice into a blender or food processor. Add the sugar-ginger mixture (including any liquid) and blitz on high for 20–30 seconds until smooth and evenly puréed.

Strain purée

Pour the purée through a fine-mesh sieve into a bowl, using the back of a spoon to press through as much liquid as possible. Discard seeds and solids left in the sieve. Taste the strained purée and adjust sweetness or lemon if needed (add up to 1 tsp more sugar or 1/2 tsp lemon juice). Chill the purée for 10 minutes if it has warmed while blending.

Finish & Serve

Combine purée and apple juice — Measure 1 cup of the strained raspberry-ginger-lemon purée and pour it into a pitcher. Add 2 cups cold fresh apple juice and stir gently until fully combined. Taste and adjust ratio: for a brighter, more tart drink use more purée; for lighter/clearer profile add more apple juice.

Add ice and chill — Add 1 cup of ice cubes to the pitcher or individual serving glasses to cool the mixture quickly. Stir briefly to chill evenly.

Serve — Divide the sherbet mixture among 4 chilled glasses (about 1 cup / 240 ml per serving). Garnish each glass with a lemon slice, two raspberries, and two mint leaves if using. Serve immediately.

Make-ahead note — The strained purée can be made up to 24 hours ahead and kept refrigerated in a sealed container. Combine with apple juice and ice just before serving to preserve freshness and carbonation (if using sparkling apple juice).