Preheat the grill to high or light the barbeque and wait until the flames have calmed down. Brush the strawberries all over with the oil after threading them onto metal skewers (or pre-soaked wooden ones). Place on the grill or barbeque, sprinkle with half of the sugar and cook for 3-4 minutes, or until they start to char. Allow cooling before serving.
Pour the cream and remaining sugar into a bowl with the quark. Whisk everything together until it's completely smooth. Sprinkle the basil over the strawberry skewers and serve with a dollop of sour cream to dip them in.