This dish combines aroma and flavor in a unique fashion: Bowls of steaming rosemary vapor accompany the luxurious main course. The piney fragrance of fresh rosemary adds another dimension to the taste experience.
Bring 8 quarts water and vinegar to a boil in a 10 to 12-quart pot. Add 2 lobsters headfirst and cook until the shells turn bright red and lobsters are just cooked through, about 11 minutes. Using tongs, carefully remove the lobsters from the water. Repeat with the remaining 2 lobsters. Twist the claws and tails off the lobsters. Crack the claws and remove the meat.
Using kitchen shears, cut away the lobster gills. Rinse the shells under cold water. Place in a large bowl. Pour cold water over and soak 10 minutes. Drain; rinse well.
Heat 1/4 cup oil in a large pot over medium heat. Add the lobster shells, carrots, onion, fennel, tomatoes, and tarragon. Cook until the vegetables begin to soften, stirring occasionally, about 15 minutes. Add 4 quarts water and simmer 45 minutes, stirring occasionally. Strain the lobster broth into a large bowl, pressing on solids. Strain the broth again through a fine strainer into the same pot. Boil over medium-high heat until the broth is reduced to 2 cups, about 1 hour. Cook's Note: Can be made 1 day ahead. Cool slightly; chill broth and lobster meat separately.
Bring 1 quart water to a boil in tea kettle. Place 2 cups rosemary in each of 4 large bowls; place on the table.
Melt 2 tablespoons butter with the remaining 1/4 cup oil in large skillet over medium-high heat. Add the mushrooms, thyme, and garlic and saute until the mushrooms are tender, about 6 minutes; discard the thyme and garlic. Add the lobster and 1/2 cup butter to the skillet and heat through, about 3 minutes.
Meanwhile, bring the lobster broth to a simmer in a small saucepan. Pour the hot lobster broth into a blender; add 1 cup butter. Blend until the butter is melted and mixture is frothy, about 30 seconds. Divide the lobster mixture among 4 shallow bowls. Pour the butter sauce and foam over the lobster mixture and serve. Pour 1 cup boiling water into each bowl of rosemary to create vapor.
NotesUse kitchen shears to cut the lobster shells. Using kitchen shears or a heavy large knife, cut the lobster tails lengthwise in half and remove the tail meat. Cut the meat into large bite-size pieces. Place in a medium bowl, then cover and refrigerate. Reserve the claw and tail shells.