Preheat grill to medium high heat. Start by preparing the lobster. Using kitchen shears butterfly the tail by cutting down the center. Loosen the meat and pull the lobster meat upward. Salt and pepper the meat and set on a baking sheet.
In a small bowl combine melted butter, olive oil, garlic and cajun seasoning. Brush on top of the prepared lobster.
Place lobster on grill flesh side down until lightly charred about 3-4 minutes. Flip over and brush again with marinade and continue to cook for about 5 more minutes until cooked throughout.