Bring large pot of salted water to boil for pasta.

Bring a stockpot of water to boil for lobster. Add lobsters, head first; cover, return to boil and cook for 2 minutes. Using tongs, transfer lobsters to cutting board. Using kitchen shears, cut off claws and remove claws and knuckle meat. Cut off lobster tails; cut tails crosswise into 4 pieces each. Cut each lobster body lengthwise in half. Remove any coral (bright orange part) and/or tomalley (greenish part); transfer, with any juices, to a bowl.

Heat 1 tablespoon oil in a large heavy saucepan or Dutch oven over medium heat. Add shallot and thyme; cook, stirring occasionally, until softened, about 3 minutes. Add lobster body pieces (not tails or claw meat); cook, stirring occasionally, for 3 minutes.

Add cognac; light on fire with a match and allow alcohol to burn off, then add wine, lobster tail pieces, claw meat, coral, tomalley and juices, and pinch salt and pepper. Cover and cook until lobster is cooked through, about 8 minutes. Remove from heat.

Add pasta to boiling water and cook until al dente; drain. In a serving bowl, toss together pasta, lobster and sauce, remaining 3 tablespoons oil and parsley. Season with salt to taste, garnish with marjoram and serve at once.