Reserve the fresh lobster tails in your refrigerator until it’s time to grill them.
Place a small saucepan over medium heat, then add the butter. Once the butter is melted add the chopped garlic/shallot/herb mixture and it a stir.
Bring the herb butter up to a light simmer and cook for 2 to 3 minutes. Turn the heat off and let the mixture rest until it’s needed.
Use kitchen shears to cut the top of the lobster tail’s shell down the middle. Work your way from beginning to end. Next, grab a sharp knife and carefully cut into the tail meat lengthwise. Leave a little bit of it connected at the bottom, then spread the meat open like a butterfly. You might have to crack the bottom of the shell a bit to expose the meat properly.
Preheat the grill to 400 degrees. Brush the lobster meat with extra virgin olive oil, season each tail with salt and pepper, then place them flesh side down on the grill. Cook the lobster tail for 4 minutes per side. Start basting each tail with the herb butter when you flip them.
Once the meat is cooked, place the lobster tails into a serving bowl or platter and drizzle some of the remaining herbed butter on top.