chipotle chile: seeded and minced cucumber: peeled and thinly sliced lengthwise using a mandoline radishes: thinly sliced lengthwise using a mandoline
Heat a charcoal grill or set a gas grill to medium-high. (Alternatively, heat a 30cm cast-iron grill pan over medium-high.) Using a heavy cleaver, split each lobster in half lengthwise through its head and tail. Scoop out and reserve tomalley. Drizzle flesh side of lobsters with oil; grill, flesh-side down, until slightly charred, about 5 minutes. Flip lobster halves over; continue cooking until cooked through, 3–5 minutes more. Meanwhile melt butter in a 2L saucepan over medium; cook garlic and ginger until soft, 2–3 minutes. Stir in reserved lobster tomalley, the seaweed, basil, lemon juice, mustard, and chile; cook until lobster tomalley has dissolved, 1–2 minutes. Arrange cucumber and radish slices on a platter; season with salt and pepper. Arrange lobster, flesh-side up, over top; drizzle with butter.