Combine basil, garlic, half the hazelnuts, parmesan and 1/3 cup oil in a small food processor. Process until well combined. Season with salt and pepper.
Lightly grease a barbecue plate and preheat on high heat. Brush both sides of mushrooms with remaining oil and barbecue stalk side down for 2 minutes, turn and barbecue a further 1-2 minutes or until mushrooms are tender. Remove to a platter.
Scatter over bocconcini, tomatoes and remaining chopped hazelnuts. Drizzle with basil mixture and serve. Serve with barbecue sausages, chicken or fish.