The base of this soup has intense corn and lobster flavors, thanks to what would normally be thrown out: the cobs and the lobster shells.
Prepare an ice-water bath. Bring 8 cups water to a boil in a large pot. Reduce heat to medium-low. Plunge lobster into water headfirst, and simmer, covered, for 9 minutes. (Do not let water boil.) Transfer lobster to ice-water bath using tongs; reserve cooking liquid. Let stand for 10 minutes to cool.
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.
Return shells and body to pot with cooking liquid. (For added flavor, chop body with a cleaver before returning to pot.) Add reserved cobs. Simmer, covered, over medium heat for 35 minutes. Strain stock through a fine sieve into a bowl; discard solids.
Heat oil in a medium saucepan 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.
Strain soup through sieve. Set aside 1 1/2 cups corn mixture. Working in batches, puree remaining corn mixture and strained liquid in a blender until smooth. (For safety, remove cap from hole in lid, and cover with a dish towel to prevent spattering.) Strain soup through sieve, and return to pot with reserved corn and lobster meat. Cook over medium heat until warmed through. Stir in chives, 1/4 teaspoon salt, and some pepper. Serve with lemon wedges.
Lobster meat can be refrigerated overnight. Soup can be refrigerated for up to 4 days.