(If you don't want to cook a live lobster for this recipe, buy 2/3 pound cooked lobster meat from your seafood shop.)
1 (1 1/4- to 1 1/2-pound) live lobster
2 teaspoons finely chopped shallot
1 Tablespoon fresh lemon juice
1/4 teaspoon table salt
2 Tablespoons extra-virgin olive oil
1 pink or ruby-red grapefruit
1 firm ripe California avocado
1 ounce baby arugula (2 cups)
Coarse sea salt to taste (optional)
Plunge lobster headfirst into an 8-quart pot of boiling salted water. Cook covered over high heat for 6 minutes (for 1 1/4-lb lobster) or 7 minutes (for 1 1/2-lb lobster). 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.
Remove 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 the other half, wrapped tightly in plastic wrap, for another use.) Halve remaining avocado 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 (optional) and serve immediately.
Makes 2 first-course servings