Prepare the live lobster. Place the lobste on a cutting board. Push a chef's knife into the cross mark on the shell between the head and body; the lobster may still move for a few seconds. Turn the lobster on its back and split it in half lengthwise with your knife or kitchen shears. Clean out the cavity. Warning: This step is messy, so have paper towels handy. Remove the rubber bands and crack the claws with the blunt edge of your knife. This will help the claw meat cook evenly on the grill.
Preheat the grill to medium.
Melt the butter with the olive oil in a small skillet over medium heat. Add the sausage and cook until the fat begins to render, 1 to 2 minutes. Add the garlic and cook until fragrant, about 1 minute. Pour about half of the butter-oil into a bowl and keep warm; leave the sausage in the skillet. Add the panko to the skillet; cook, tossing, until crisp and golden, about 5 minutes. Stir in the parsley, chives and lemon juice and season with salt and pepper.
Brush the lobster tails on the grill shell-side down; cover with a disposable aluminum pan and cook until the shells turn red and the stuffing is crisp, 12 to 15 minutes. Sprinkle with parsley; serve with lemon wedges and the remaining butter-oil for dipping.