Make the aioli by combining the first 6 ingredients in a small bowl. Cover and refrigerate until needed. Preheat your barbecue or indoor grill to medium-high. Cut each lobster tail lengthwise in half. Use paper towel to remove any excess moisture on the lobster. Brush the lobster with the oil; season with salt and pepper. Lightly oil the bars of your grill. Grill the lobster, cut side down, for 2 minutes. Turn cut side up and grill 1 to 2 minutes more, or just until cooked through. When cooked, the lobster flesh should feel just slightly firm, not rubbery, a sign it’s overcooked.
Recipe Options: Instead of aioli, simply serve the lobster with melted butter and wedges of lemon. Or serve the lobster with champagne hollandaise sauce for dipping. You'll find a recipe for that sauce in this website collection recipe: Steak and Lobster with Champagne Hollandaise.