Step 1: Plunge lobster headfirst into an 8-quart pot of boiling salted water . Cook, covered, over high heat 6 minutes (for 1 1/4-lb lobster) or 7 minutes (for 1 1/2-lb lobster) from time it enters water. Transfer with tongs to sink to drain.

Step 2: When lobster is cool enough to handle, remove meat from tail and claws, keeping meat intact. Discard tomalley, any roe, and shells. Chill lobster, covered, until cold, at least 1 hour.

Step 3: While lobster chills, stir together shallot, lemon juice, and table salt in a small bowl and let stand at room temperature 30 minutes. Add oil in a stream, whisking.

Step 4: Cut peel, including all white pith, from grapefruit with a sharp knife. Cut segments free from membranes and transfer segments to paper towels to drain.

Step 5: Halve avocado lengthwise, discarding pit. (Save 1 half, wrapped tightly in plastic wrap, for another use.) Halve remaining avocado half lengthwise and peel, then cut crosswise into 1/3-inch-thick slices.

Step 6: Cut lobster tail meat crosswise into 1/2-inch-thick slices. Divide avocado and all of lobster meat between 2 salad plates and arrange grapefruit around them. Top with arugula and drizzle with dressing. Sprinkle lightly with sea salt (if using) and serve immediately.