If using live lobsters, bring a large pot of water to a boil. Plunge the lobsters in head first and cook over high heat until the lobsters are red almost all over, about 5 minutes. The water may just have returned to a boil at this point. Using tongs, transfer the lobsters to a large bowl and let cool.
When the lobsters are cool enough to handle, separate the heads from the bodies. Using kitchen shears, snip the underside of the tail shells down the center and remove the lobster tail meat in 1 piece. Remove the dark intestinal vein in the tail, then slice the meat 1/4 inch thick. Twist off the claws, crack them with a mallet and remove the meat. Cut the meat from each claw into 3 pieces. Cover and refrigerate the lobster meat if not making the recipe straight through. Reserve the lobster shells, including the heads.
Rinse the insides of the lobster shells, then break them up into very small pieces. In a large saucepan, heat the pure olive oil. Add the lobster shells and cook over moderately high heat, stirring often, until the shells start to brown, about 5 minutes. Add the shallots and thyme sprigs and cook, stirring, until the shallots start to brown, about 3 minutes. Reduce the heat to moderate and add the tomato paste. Cook, stirring, until the tomato paste starts to stick to the bottom of the pan, about 3 minutes. Add the Cognac and simmer, scraping up the browned bits from the bottom. Add the water and a pinch of salt and bring to a simmer. Cook the lobster stock over low heat until reduced to 2 cups, about 20 minutes.
Skim the lobster oil from the surface and discard. Line a fine strainer with 4 layers of dampened cheesecloth and slowly strain the lobster stock through it into a clean large saucepan.
Put the porcini in a heatproof bowl. Bring 2 cups of chicken stock to a simmer and pour over the porcini; let the mushrooms soften for 20 minutes. Rub the porcini to loosen any grit, then lift them out. Reserve the porcini for another use. Let the soaking liquid stand for 5 minutes so the grit falls to the bottom, then pour the liquid into the pan with the lobster stock, leaving behind any grit. Add the remaining 6 cups of chicken stock to the saucepan.
Bring the broth to a bare simmer. Season with salt and white pepper and keep warm over very low heat.
Bring the lobster meat to room temperature if needed and season lightly with salt. Spoon the lobster into 10 warmed shallow soup plates. Ladle the broth over the lobster meat and garnish with a few chervil leaves. Drizzle with extra-virgin olive oil and serve.
The recipe can be prepared through Step 5 up to 1 day ahead. Refrigerate the broth, lobster meat and lobster oil separately.
A fresh, aromatic Australian Riesling is an elegant contrast to the mild soup.