Fill a large bowl with ice and water; set aside. Bring 2 inches of water to a boil in a large stockpot. Steam lobsters for 10 minutes; transfer to ice bath until cool. Break off claws and tails; use bodies to make a stock, or discard. Split tail lengthwise with a sharp knife; remove meat. Slice tail into bite-size pieces. Remove claws from shells, keeping whole if possible. Chill lobster meat.
Cut ends off grapefruit; remove peel, pith, and outer membranes. Lift sections away from the membranes, and reserve. Squeeze 1/2 cup juice from the membranes into a bowl for dressing.
Place shallots, mustard, vinegar, juice, and lemon basil in a bowl; season with salt and pepper. Whisk to combine. Gradually whisk in olive oil; adjust seasoning.
Cut avocados into 1/2-inch-thick wedges; coat with a little grapefruit juice to prevent discoloration. Add lobster, grapefruit, and hearts of palm. Line dishes with Bibb lettuce leaves. Tear remaining lettuce into bite-size pieces; add to lobster mixture. Drizzle with dressing; toss. Fill lettuce leaves with salad; garnish with lemon basil.
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