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Fill a large bowl with ice and water; set aside. Bring a large pot of water to a boil over high. Add lobster to water headfirst, and cook until lobster is bright red and a meat thermometer inserted in center of tail registers 140°F, about 10 minutes. Transfer lobster to ice bath, and let cool about 10 minutes.
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Meanwhile, fill a separate large pot with 1 inch of water. Place an expandable metal steamer basket in pot, and bring water to a boil over medium-high. Reduce heat to medium to maintain a simmer. Add scallops to steamer basket; cover and cook until white, opaque, and slightly firm, 2 to 3 minutes. Transfer to a large plate. Add squid to steamer basket; cover and cook until tender, 1 minute for tentacles and 3 minutes for tubes. Transfer to plate with scallops. Add mussels to steamer basket; cover and cook until shells open, about 3 minutes. Transfer mussels to a medium bowl, discarding any mussels that have not opened. Refrigerate seafood, uncovered, until thoroughly chilled, about 30 minutes.
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Cut lobster tail lengthwise through the top of shell using kitchen shears. Using your thumbs and fingers, gently spread the shell halves apart, separate the meat from the shell, and discard shell. To remove the meat from the claw, wiggle smaller hinged portion of each claw to separate. Break open claws using a mallet or lobster-cracking tool, cracking one side and then flipping to crack other side, and remove meat. Cut claw and tail meat crosswise into 1/2-inch-thick slices. Place claw meat in a large bowl and tail meat in a small bowl.
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Slice scallops horizontally into 1/4-inch-thick rounds. Cut squid tubes crosswise into 1/4-inch rings, leaving tentacles whole. Remove mussels from shells, and discard shells. Add scallops, squid, and mussels to lobster claw meat in bowl. Stir in shallot, basil, parsley, 1/2 tablespoon chives, and 1/2 cup vinaigrette. Sprinkle with remaining 3/4 teaspoon salt. Season with additional white pepper to taste. Cover and refrigerate scallop mixture and lobster tail meat at least 15 minutes or up to 1 hour.
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Add frisée and oakleaf lettuce to scallop mixture, and toss gently to coat. Mound 1 heaping cup scallop mixture in center of 4 large plates. Peel and quarter avocado; thinly slice each quarter lengthwise. Lightly brush avocado with vinaigrette. Toss lobster tail meat with 2 tablespoons vinaigrette, and arrange 2 to 3 slices on one side of scallop mixture. Arrange one-fourth of avocado slices on opposite side.
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Garnish salads with lemon peel strips, if using, and remaining 1/2 tablespoon chives. Serve remaining vinaigrette on the side.