- 1 cup distilled white vinegar
- 4 1 1/2-pound live Maine lobsters
- 1/2 cup canola oil
- 3 carrots, peeled, thinly sliced
- 1 onion, thinly sliced
- 1 fresh fennel bulb, thinly sliced
- 2 tomatoes, thinly sliced
- 4 fresh tarragon sprigs
- 8 cups fresh rosemary leaves stripped from branches (about ten .66-ounce packages)
- 2 tablespoons plus 1 1/2 cups (3 1/4 sticks) butter, room temperature
- 2 pounds assorted fresh wild mushrooms (such as oyster, crimini, chanterelle, and stemmed shiitake), cut into 1-inch pieces
- 4 large fresh thyme sprigs
- 3 garlic cloves, halved
- Bring 8 quarts water and vinegar to boil in 10- to 12-quart pot. Add 2 lobsters headfirst and cook until shells turn bright red and lobsters are just cooked through, about 11 minutes. Using tongs, remove lobsters from water. Repeat with remaining 2 lobsters. Twist claws and tails off lobsters. Crack claws and remove meat. Using kitchen shears or heavy large knife, cut lobster tails lengthwise in half; remove tail meat. Cut meat into large bite-size pieces. Place in medium bowl; cover and refrigerate. Reserve claw and tail shells.
- Using kitchen shears, cut away lobster gills. Rinse shells under cold water. Place in large bowl. Pour cold water over and soak 10 minutes. Drain; rinse well.
- Heat 1/4 cup oil in large pot over medium heat. Add lobster shells, carrots, and next 4 ingredients. Cook until vegetables begin to soften, stirring occasionally, about 15 minutes. Add 4 quarts water and simmer 45 minutes, stirring occasionally. Strain lobster broth into large bowl, pressing on solids. Strain broth again through fine strainer into same pot. Boil over medium-high heat until broth is reduced to 2 cups, about 1 hour. (Can be made 1 day ahead. Cool slightly; chill broth and lobster meat separately.)
- Bring 1 quart water to boil in tea kettle. Place 2 cups rosemary in each of 4 large bowls; place on table.
- Melt 2 tablespoons butter with 1/4 cup oil in large skillet over medium-high heat. Add mushrooms, thyme, and garlic and sauté until mushrooms are tender, about 6 minutes; discard thyme and garlic. Add lobster and 1/2 cup butter to skillet and heat through, about 3 minutes.
- Meanwhile, bring lobster broth to simmer in small saucepan. Pour hot lobster broth into blender; add 1 cup butter. Blend until butter is melted and mixture is frothy, about 30 seconds. Divide lobster mixture among 4 shallow bowls. Pour butter sauce and foam over lobster mixture and serve. Pour 1 cup boiling water into each bowl of rosemary to create vapor.