No melted-butter baths here. Instead, succulent sweet lobster is tossed with Ruby Red grapefruit dressing, a pleasingly tart counterpoint with a hint of mint.
Preheat broiler, with rack as close as possible to heat source. Cut ends off grapefruits. Using a paring knife, remove peel, pith, and outer membranes, following the curve of the fruit. Slice between membranes to release sections, and transfer to a small bowl. Squeeze juice from membranes into a separate bowl, and reserve (you should have about 1/4 cup juice).
Using kitchen shears, cut along length of each lobster-tail shell. Using a knife, make a shallow cut in the meat, leaving tail intact. Rinse, and drain. Using your fingers, loosen meat from shells but do not remove.
Arrange lobster tails and grapefruit segments on a rimmed baking sheet in a single layer. Rub lobster meat with oil, salt, and pepper. Drizzle reserved grapefruit juice on top.
Broil until lobster tails are opaque and edges of grapefruits are browned, 5 to 6 minutes. (The shells will get dark, so watch the meat closely to ensure it doesn't overcook.) Let lobster meat cool slightly, about 10 minutes. Remove meat from shells, and discard shells. Cut lobster meat in half lengthwise, then into 1/2-inch diagonal slices.
Toss lettuce, endive, and chives in a large serving bowl.
Make the vinaigrette: Whisk all the ingredients in a small bowl.
Add lobster meat and vinaigrette to lettuce mixture, and toss to coat. Gently fold in broiled grapefruit segments. Serve immediately.
Frozen lobster tails are a fine substitute for fresh, but thaw them completely before broiling.