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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.