Step 1

Bring stock (or broth) and water to a boil in a large stockpot.

Step 2

Very carefully remove the rubber bands from lobsters (pay attention; they will start trying to pinch). Place the lobsters claws-side down in the pot, cover and cook for 5 minutes. Uncover, carefully flip the lobsters over, cover and cook for 5 minutes more. Remove the lobsters to a large bowl and let cool. (Reserve the stock.)

Step 3

Working over a large bowl to catch the lobster juice, cut open the tails with kitchen shears. Insert a fork into a tail on an angle and pull out the meat. Crack open the claws with a lobster cracker or nutcracker and remove the meat from the claws and knuckles. (The meat will not be fully cooked.) Cut the meat into 1-inch pieces; set aside in the bowl. Remove the thin legs from the body, break in half and add to the stock.

Step 4

Meanwhile, heat butter and oil in a large pot over medium-low heat. Add leeks and cook, stirring occasionally, until very soft but not browned, about 10 minutes. Add potatoes and cook for 5 minutes. Increase heat to high and add the reserved stock (with the legs) and any reserved lobster juice; bring to a boil. Reduce heat to maintain a gentle simmer, cover and cook until the potatoes are almost tender, 10 to 12 minutes more.

Step 5

If using fresh corn, cut the ends so you can stand the cob flat on a work surface. Using a large, sharp knife and standing each cob on its end in a large shallow dish, cut the kernels from the cob. Then, using the blunt side of the knife, scrape down the cob to release the corn "milk." Repeat with the remaining ears of corn.

Step 6

Stir the corn kernels and corn milk (or frozen corn), reserved lobster meat and its juice, cream (or half-and-half), 1/4 cup chives, pepper and salt into the pot. Cook over medium-low heat until steaming hot and gently simmering, 5 to 6 minutes. Serve topped with the remaining 1/4 cup chives.