Step 1: Preheat the grill to high.
Step 2: If using lobster tails, cut them in half lengthwise with kitchen scissors or a sharp, heavy knife; use a fork to remove the intestinal vein running the length of the tail. If using live lobsters, kill each by inserting a sharp knife in the back of the head between the eyes; this will dispatch them instantly. (Alternatively, bring 8 quarts of water to a boil in a large pot with a lid. Add the lobsters, cover the pot tightly, and boil for 2 minutes. Using tongs, transfer the lobsters to a platter and let cool. Then cut in half and grill as directed.)
Step 3: Cut the lobsters in half lengthwise and remove the vein and the papery gray sac from the head. Break off the claws and crack with a nutcracker. Set the lobsters or tails aside while you prepare the ginger-mint butter.
Step 4: Melt the butter in a small, heavy saucepan over low heat. Add the mint, ginger, garlic, and lime zest and increase the heat to medium. Cook until the mixture is fragrant, but not brown, about 3 minutes. Stir in the fish sauce and lime juice and bring to a boil; then remove from the heat.
Step 5: Brush the cut sides of the lobster tails or lobsters with the ginger-mint butter and sprinkle with salt and pepper.
Step 6: When ready to cook, oil the grill grate. Arrange the lobster halves or tails and claws (if any) cut sides down on the hot grate and grill for 6 to 8 minutes. Turn, using tongs, and grill on the shell sides until the flesh is firm and white, 6 to 8 minutes more, brushing generously several times with the butter as the lobsters cook.
Step 7: Transfer the lobsters to serving plates or a platter and pour any remaining butter over them. Serve immediately.