Boil down sugar and water to a reduction (sugar syrup) (takes about 10 minutes), then let them cool in a water bath. Clean, wash and dry the basil leaves. Halve and squeeze the lemon. Remove the white skin from one half of the peel, first cut into fine strips, then into fine cubes. Chop the pepper in a mortar (not in an electric chopper!).
Finely puree the basil with the sugar syrup (best in a blender), add lemon juice to taste. Add yoghurt, lemon zest and pepper, stir to a homogeneous mass.
Fill into a flat bowl (with lid) and place in the freezer for at least four hours (stir every 30 minutes at the beginning) or prepare in the ice cream maker.