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.
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.
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.
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.
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.
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.