Set a large 2 to 3 gallon stockpot to boil over high heat. Once the water comes to a boil, immerse the lobsters in the pot. Cook the lobsters for 3 minutes, remove, and chill immediately in an ice bath. Once the lobsters are cool enough to handle, remove from the water and place on a kitchen towel draped over a cutting board.

Twist the tails off the lobsters, and cut the tails in half lengthwise using a large chef's knife (score along the length of the tail). Set the cut tails aside, and crack the shells of the claws to remove the meat. Reserve the claw meat for pasta or a salad for a later date.

Place the Jerusalem artichokes in a 1-quart saucepan, and cover with water. Set over high heat and bring to a boil. Once the water comes to a boil, cook the artichokes for 15 minutes, or until easily pierced with a knife. Remove from the heat, drain, and place the artichokes in a food processor with the heavy cream, 1 tablespoon of butter, 1/4 teaspoon of salt and 1/8 teaspoon of white pepper. Process the artichokes for 10 to 15 seconds, or until a smooth puree is formed. Remove from the bowl of the food processor, and place in a heat resistant bowl set over a water bath to keep warm.

Trim the ends off of the asparagus and blanch in boiling water for 2 minutes. After 2 minutes, remove from the water and cool in ice water. Once the asparagus is cool enough to handle, remove from the ice water and pat dry with a paper towel. Cut the asparagus into thirds on a bias.

Set a small saucepan over a low heat and add 4 tablespoons of the remaining butter, the lemon zest, and the lemon juice. Bring to a gentle simmer and cook for 3 to 4 minutes, allowing the flavors to meld, and season with a pinch of salt, white pepper, and the chopped tarragon. Set aside, but keep warm.

To finish the dish, set the grill on medium-high. Brush the lobster tails with the olive oil on the cut end, and season with the remaining 1/4 teaspoon of salt and 1/8 teaspoon of white pepper. Place the tails, cut-side down, on the grill, and cook for 3 minutes. Rotate 45 degrees, and cook for an additional 3 minutes. Turn the lobsters over and cook for 2 minutes. While the lobsters are cooking on the shell side, place a small skillet over medium-high heat, and once the pan is hot, add the remaining 1 teaspoon of butter and saute the asparagus, tossing, for 1 minute. Season lightly with salt and white pepper.

To plate the dish, divide the Jerusalem artichoke puree evenly between 2 warmed dinner plates or wide shallow bowls. Arrange the asparagus around the artichoke puree, and place the lobster tails on top of the puree. Pour 2 tablespoons of the lemon- tarragon butter over each serving of lobster and garnish with additional tarragon if desired.

Serve immediately.