Mash all the butter ingredients together. Set aside half for serving; melt the other half for basting.
Bring a large pot of salted water to a boil. Plunge the lobsters in head first for 3 minutes, then transfer to an ice bath. This makes them easier to split and keeps the meat from overcooking on the grill.
Place each lobster belly-down and split lengthwise with a heavy knife from head to tail. Remove the stomach sac and intestinal vein. Crack each claw once with the back of the knife.
Heat the grill to medium-high and oil the grates. Brush the cut sides with olive oil and grill flesh-side down 3–4 minutes until lightly charred. Flip shell-side down, spoon melted garlic butter over the meat and grill 4–5 minutes more, basting, until the tail meat is opaque throughout. Put the claws on alongside for the full time.
Top with spoonfuls of the reserved butter and serve with lemon wedges. Corn on the cob and a cold rosé finish the job.
