In a large stockpot over medium-high heat, bring the fish stock to a vigorous simmer. Very carefully remove the rubber bands from the lobster claws (the lobsters can pinch you, so pay attention). Place the lobsters, shell side up, in the pot, cover, and cook for 5 minutes. Remove the lid, flip the lobsters, and cook for another 5 minutes. Turn off the heat. Using tongs, remove the lobsters and let cool. Working over a flat, rimmed dish, such as a pie plate, to catch any released juice, remove the meat from the claws and the tail and cut into generous bite-size pieces; reserve the lobster meat and juice. Remove the legs (not the claws but the thin spiny legs on the body), cut in half and reserve.
Trim off the dark green sections from the leeks and save for making vegetable stock. Halve the pale green and white sections. Rinse under cold running water, pat dry, and cut crosswise into thin pieces.
In another large stockpot over low heat, melt the butter. Add the leeks and cook, stirring occasionally, for 5 minutes. Add the potatoes and half of the chives and cook, stirring, for 2 minutes. Turn the heat to high, add the stock and reserved lobster juice, and bring to a boil. Turn the heat to low, cover, and cook for 10 to 12 minutes, or until the potatoes are just tender.
In a small nonreactive saucepan over low heat, bring the cream to a gentle simmer. Remove from the heat, add the lobster pieces and the leg pieces and let steep for 3 minutes. Then, add the cream and lobster to the stock.
If using fresh corn, shuck the ears, remove the silks, and trim off the ends so that you can stand the cob flat. Using a sharp knife and standing each cob on its end inside a large bowl, remove the kernels from the cob by working the knife straight down against the cob. Using the blunt side of the knife, scrape down the cob after the kernels have been removed to release the corn “milk,” Repeat with the remaining ears. Add the corn kernels, corn milk, and 2 tablespoons chives to the pot; season with salt and pepper; and cook for 5 minutes, or until the soup simmers. Taste and adjust the seasoning, adding more salt and pepper if necessary.
Ladle the chowder into mugs or bowls and garnish with the remaining 2 tablespoons chives and the paprika before serving.