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1
Bring water to a boil in pot, then plunge 2 lobsters headfirst into water and boil, covered, 8 minutes from time they enter water.
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2
Transfer with tongs to a shallow roasting pan to cool.
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3
Cook remaining 2 lobsters in same manner, reserving cooking liquid in pot.
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4
When lobsters are cool enough to handle, remove meat from tail and claws, catching juices in pan.
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5
Scrape off any coagulated white albumen from meat with point of a small knife, then cut meat into 1/2-inch pieces and chill in a bowl, covered.
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6
Cut lobster bodies in half lengthwise, then transfer to cooking liquid along with shells, tomalley, and any juices from roasting pan.
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7
Heat butter in a 12-inch heavy skillet over moderately high heat until foam subsides, then stir in carrots, celery, fennel, garlic, tarragon, and salt.
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8
Reduce heat to moderately low and cook vegetables, covered, stirring occasionally, until tender but not browned, about 15 minutes.
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9
Remove lid and add wine, then boil over high heat until most of liquid is evaporated, about 7 minutes.
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10
Add vegetable mixture to lobster-shell mixture and bring to a boil, skimming foam.
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11
Boil, uncovered, until stock is reduced to about 8 cups, 1 to 1 1/2 hours.
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12
Discard lobster bodies, then pour stock through a cheesecloth-lined sieve into a large bowl, pressing on and then discarding solids.
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13
Set aside 3 cups stock for sauce and reserve remaining stock for another use (cool stock, uncovered, then chill, covered, or freeze).
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14
Cut an X in end opposite stem of tomatoes and immerse in a pot of boiling water 10 seconds.
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15
Transfer tomatoes with a slotted spoon to a bowl of ice and cold water to stop cooking, reserving boiling water.
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16
Peel and seed tomatoes (reserve ice water) and cut into 2- by 1/4-inch strips.
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17
Add kosher salt to boiling water, then add carrots and cook until just tender, about 5 minutes.
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18
Transfer with slotted spoon to bowl of ice water.
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19
Cook haricots verts and peas in boiling water until crisp-tender, about 4 minutes, then transfer with slotted spoon to ice water.
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20
Drain vegetables in a colander and pat dry.
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21
Cook shallot in 3 tablespoons butter in a 4- to 5-quart heavy pot over moderate heat, stirring occasionally, until shallot is softened, 4 to 5 minutes.
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22
Add fennel and salt and pepper to taste, then cook, stirring occasionally, until fennel is golden and tender, about 15 minutes.
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23
Add 2 3/4 cups lobster stock to fennel along with tomato strips and bring to a simmer.
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24
Whisk together cornstarch and remaining 1/4 cup stock in a cup until smooth, then whisk into sauce.
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25
Simmer sauce, whisking, until slightly thickened, about 2 minutes, then whisk in remaining 3 tablespoons butter until incorporated.
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26
Stir in lemon juice and salt and pepper to taste.
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27
Add blanched vegetables, lobster, and tarragon and gently simmer, stirring occasionally, just until heated through, 4 to 5 minutes.
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28
Divide stew among 6 bowls and top with potato rosettes, upside down.
