Cook all the vegetables (except the mushrooms) separately in salted boiling water until crisp-tender and then chill them. Saute the mushrooms without butter for a few minutes until tender. Set all the vegetables aside until almost ready to serve.
Place the lobsters in a tight-fitting heatproof container. Cover with cold water. Drain off the water, measure it, and place the water into a large pot. Bring the water to a boil and add 1/2 cup distilled white vinegar per 8 quarts water. Pour the boiling liquid over the lobsters and let them steep for 2 minutes if using 1 1/2-pound lobsters and 3 minutes if using 2-pound lobsters. Remove the lobsters from the hot water.
Using a towel or rubber gloves, hold the lobster body, and remove the lobster tail by twisting it and detaching from the body by lightly pulling on it. Remove the claws the same way, and place the claws back into the hot water. Set the body on a plate. Repeat with the other lobster.
Hold each piece of tail meat flat and twist the tail fan from one side, pull off and discard. Using your fingers, gently push the meat through the tail and pull the meat out through the large opening at the other end. Discard the shell or keep it for stocks and bisques. Lay the tail meat on the rounded side and halve it lengthwise through the middle. Remove the vein that runs through the top of the meat. Lay the meat on a paper towel-lined plate, cover with plastic wrap, and refrigerate.
Remove the claws from the water. Twist off each knuckle and reserve. Hold each claw and pull down on the smaller portion of the claw to loosen it and remove it from the larger part of the claw, keeping the meat intact and attached. Still holding the claw, crack the top of the shell with the heel of the knife where the knuckle was attached. Go through the shell but not deep enough to damage the meat. Wiggle the knife to crack the shell. If the shell does not pop off, it may be necessary to turn the claw over and repeat the procedure. Shake the claw meat to remove the meat (if it doesn't fall out, cut off the very tip of the shell and blow through the hole to release the meat). Place claw meat on the plate with the tail meat. Repeat with remaining claws.
Cut off the top joint of each knuckle, the one that was attached to the lobster's body. Use scissors to cut away the shell along the smooth outside edge of the knuckle. Use your fingers to pry open the shell and remove the meat. Place the knuckle meat on the plate with the tail meat.
For the lobsters, bring the lobsters to room temperature. Place them in 1 layer in a large saucepan with the melted butter. Over low heat, poach the lobster meat for 5 to 6 minutes.
Slice the lobster pieces into thick slices all about the same size. In a saucepan, heat 1 teaspoon of the poaching butter and add the shallots. Add the vinegar and reduce to almost dry. Add the wine and reduce by 3/4. Add the cream and reduce by 2/3. Add the lobster pieces to the pan and saute for 1 minute. Add the Cognac to the pan and keep the pan warm. Adjust seasonings, if necessary.
In a large saute pan, heat 1 tablespoon of the poaching butter. Saute the cooked vegetables with salt and pepper until just heated through and divide the vegetables evenly between 4 plates.
Remove the lobster pieces from the Cognac sauce and plate on top of the vegetables, divided evenly. Strain the sauce over the lobster and vegetables. Garnish with basil and chives.
Recipe courtesy of Thierry Burkle, L'Etoile, San Antonio, Texas