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.