How to Make It
Bring a large pot of water to a boil. Plunge the lobsters in head first, then cover and cook until they are bright red all over, about 8 minutes. Transfer the lobsters to a large rimmed baking sheet and let cool slightly. Twist off the tails and claws. Crack the shells and remove the meat from the tails, claws and knuckles. Halve the tail meat lengthwise and discard the intestinal veins.
In a small saucepan, combine the kimchi and kimchi liquid, rice vinegar and shallot and boil over moderately high heat until the liquid has reduced to 3 tablespoons, about 3 minutes. Remove from the heat and whisk in 6 tablespoons of the butter, 1 tablespoon at a time, to form a smooth, creamy sauce. Cover and set aside.
In a large skillet, toast the crushed cardamom seeds and the mustard seeds over moderately high heat until fragrant, about 30 seconds. Add the remaining 4 tablespoons of butter and remove from the heat. When the butter has melted, add the lobster meat and stir to coat with butter. Cover and set the skillet over low heat, turning the lobster a few times, until just heated through, about 3 minutes.
Very gently rewarm the kimchi butter sauce over low heat, whisking constantly. Whisk in the tarragon and season with salt and pepper.
Transfer the lobster to plates, spoon the kimchi butter sauce on top and serve.