1. In a blender or food processor combine butter and basil. Blend until smooth. Season with salt and pepper to taste and blend to mix. If you are using salted butter you might not need to add salt. Transfer the mixture into a small bowl, cover and refrigerate.
2. Prepare lobster tails by splitting in half. To do this use a large, sharp knife. Place the lobster tails on a cutting board backside down.
Cut down through the center to the shell the whole length. Fold back, breaking through the shell. You can use kitchen shears to cut through the shell if you need to. Brush flesh lightly with oil and place cut side down on a preheated grill over a medium heat. Grill for 5 to 7 minutes depending on the size of the lobster tails.
3. Remove from grill, brush heavily with basil butter, and serve.