Preheat a charcoal grill for direct grilling.
Split each lobster tail down the underside with a heavy knife, taking care not to cut through the back shell, so that they are still in one piece that you can open like a book. Brush the lobster meat with some of the oil and sprinkle with salt and pepper.
Place the lobsters cut-side down on the grill and cook until lightly charred and heated through, about 4 minutes. Flip and heat briefly, another 30 seconds or so. Remove from the shell, being careful to leave the meat from each half in one piece.
Spread some of the Charred Corn and Avocado Salsa on the bottom of the hamburger buns. Top with 2 pieces lobster, some sliced red onions and a squeeze of lime juice. Spread extra mayo on the top bun, if desired, top the sandwiches and serve.