Cook lobsters: 2 lobsters headfirst into boiling water, cook, covered, 8 min. from time
they enter water. Transfer to roasting pan; cool. Cook remaining 2 as above,
reserving Remove meat, catching juices in pan. Cut meat into 1/2-inch pieces and
chill, covered.
Cut lobster bodies in half lengthwise, then transfer to cooking liquid along
with shells, tomalley, and juices from roasting pan.
Stock:
Heat butter or oil in a heavy skillet; med. high heat until foam subsides; add
carrots, celery, fennel, garlic, tarragon, salt. Reduce, cook vegs. covered, stirring occasionally, until tender but not browned, about
15 minutes. Remove lid and add wine, then boil over high heat until most of
liquid is evaporated, about 7 minutes. Add vegetable mixture to lobster-shell
mixture and bring to a boil, skimming foam. Boil, uncovered, until reduced to
about 8 cups, 1 to 1 1/2 hours.
Discard lobster bodies; pour stock through a sieve to a large bowl, pressing
on and then discarding solids.
Prepare vegetables for stew:
Cut an X in end opposite stem of tomatoes and immerse in a pot of boiling water
10 seconds. Transfer tomatoes to ice water. Peel/seed
tomatoes; cut into 2- by 1/4-inch strips.
Add salt to boiling water, cook carrots until just tender, about 5
minutes. Transfer to ice water. Cook haricots verts and peas until crisp,
about 4 minutes, then ice water. Drain and pat dry the vegs.
Make sauce:
Cook shallot in 3 tbsp butter in a 4- to 5-quart heavy pot over moderate heat,
stirring occasionally, until shallot is softened, 4 to 5 minutes. Add fennel,
salt and pepper, until fennel is golden and tender, about 15 minutes.
Add 7 ½ cups lobster stock to fennel along with tomato strips and bring to a
simmer. Whisk cornstarch and remaining ½ cup stock til smooth, then whisk
into sauce. Simmer, whisking, until slightly thickened, about 2 minutes, then
whisk in remaining 3 tbsp butter until incorporated. Stir in lemon juice, salt
and pepper to taste. Add blanched vegetables, lobster, and tarragon and gently
simmer, stirring occasionally, just until heated through, 4 to 5 minutes.