Grilled Lobster Salad with Avocado and Papaya

Prepare barbecue (medium-high heat). Place smoke chips in 8 x 6-inch foil packet with open top. Set atop coals 5 minutes before grilling lobsters. Bring large pot of salted water to boil. Working in batches, drop lobsters headfirst into water. Cover; boil 2 minutes. Transfer lobsters to work surface. Immediately split in half lengthwise, using heavy large knife. Crack claws. Arrange lobsters, cut side up, on grill. Cover; grill 6 minutes. Turn lobsters over and grill until just cooked through, about 5 minutes. Remove from heat. Remove lobster meat from shells. Cool. Cut meat into 1/2-inch pieces. Transfer to large bowl. Whisk oil and next 5 ingredients in medium bowl to blend. Season with salt and pepper. (Can be made 1 day ahead. Cover lobster and dressing separately; chill. Rewhisk dressing before continuing.) Add avocados, tomatoes and papaya to lobster in bowl. Pour 1/2 cup dressing over; toss. Season with salt and pepper. Toss greens in another large bowl with enough remaining dressing to coat. Divide greens among plates. Spoon lobster mixture atop greens and serve.