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1. Bring 8 cups water to a boil in a large, deep pot. Reduce heat to medium-low. Plunge lobster into water head first and simmer, covered, taking care to not let the water boil. Meanwhile, prepare an ice water bath. After 9 minutes, transfer lobster to the ice water bath using tongs; reserve cooking liquid. Let lobster stand for 10 minutes to cool.
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2. Crack lobster claws, knuckles, and tail, and remove meat; reserve shells and body. Coarsely chop meat (you should have about 1 cup). Refrigerate until ready to use.
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3. Return shells and body to the pot with cooking liquid. Add reserved cobs and simmer, covered, over medium heat for 35 minutes. Strain stock through a fine sieve into a bowl; discard solids.
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4. Add oil to the same pot over medium heat. Cook corn kernels, onion, garlic, and 3/4 teaspoon salt, covered, stirring occasionally, until tender, about 20 minutes. Add 5 cups lobster stock, and cook for 15 minutes. Let cool slightly.
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5. Use a slotted spoon to strain about 1 1/2 cups corn mixture; set aside. Working in batches, puree remaining corn/stock mixture in a blender until smooth. (For safety, remove cap from the hole in the lid, and cover with a dish towel to prevent spattering.)
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6. Strain soup through a sieve back into the pot, using the back of a ladle to push all liquid through. Add reserved corn and lobster meat, cooking over medium heat until warmed through. Stir in chives, 1/4 teaspoon salt, and some pepper. Serve immediately with lemon wedges.
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Make ahead: Lobster meat can be refrigerated overnight.
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Storage: Soup can be refrigerated for up to 4 days.
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Nutritional info per serving (about 1 1/4 cups): 229 calories, 1g saturated fat, 5 g unsaturated fat, 27g carbohydrate, 19g protein, 4g fiber