On a side burner or stove top, heat the butter, garlic, Worcestershire sauce, hot chilli sauce, lemon juice and smoked paprika in a saucepan over high heat. Leave the mixture to simmer for 2 to 3 minutes.
Prepare the barbecue for indirect cooking over medium heat (190°C to 230°C). If using a Weber® Q® barbecue, set up your barbecue with a convection tray and a trivet.
Place a teaspoon of the butter mixture over each oyster and top with a sprinkle of the grated Parmesan cheese.
Carefully place the oysters on the barbecue and roast over indirect medium heat, with the lid closed, until the juices have started to bubble, about 5 to 10 minutes.
Remove the oysters from the barbecue and garnish with a sprinkle of the finely chopped chives.