Preparation 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.
Preparation 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.