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.
Charred Corn-Avocado Salsa:
Preheat the grill to medium direct heat.
Pull the outer husks down the corn to the base. Strip away the silk from each ear of corn by hand. Fold the husks back into place and tie the ends together with kitchen string. Place the ears of corn in a large bowl of cold water with the salt for 1 hour.
Remove the corn from the water and shake off any excess water. Place the corn on the grill, close the cover and grill until the kernels are tender, 15 to 20 minutes, turning every 5 minutes. Let cool slightly. Remove the kernels from the ears.
Put the corn kernels, creme fraiche, cilantro, avocados, lime juice, chiles, onions and hot sauce in a medium bowl and toss gently, breaking up the avocado a bit. Season with salt and pepper.