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Place the crabmeat, chives and oil in a medium bowl and toss gently. Season with salt and pepper and refrigerate until chilled.
Cut the avocado in half lengthwise and remove the seed. Scoop the entire halves of the avocado flesh from the shells in one piece and place on a cutting board, cut sides down. Squeeze 1/2 of the lemon over the piece. Cut the avocado lengthwise into 1/4-inch-thick slices.
Have 6 chilled salad plates ready. Place 3 slices of avocado in the center of each. Place equal portions of chilled citrus mixture on top of the avocado, along with any liquid that has accumulated. Top with crab mixture and garnish with greens. Sprinkle with a little of the juice from the remaining lemon half and serve immediately.