Cut the vanilla bean in half and split lengthwise. Place in a small saucepan with the thyme, shallots, and champagne. Bring just to a boil, reduce the heat, and simmer until reduced to ¼ cup. The reduction will be a clear golden color.
Remove from the heat and strain the reduction through a sieve. Reserve the vanilla bean. Rinse the saucepan and return the reduction to the pan. Using a sharp paring knife, scrape the seeds from the vanilla bean into the reduction.
Bring the reduction to a simmer and whisk in the butter, 1 tablespoon at a time to fully incorporate, until glistening and smooth. Adjust the seasoning with salt and pepper, if necessary. Remove from the heat and set aside until the lobsters are ready. Keep warm.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Place the tails, shell-side down, on a cutting board. Split the tails lengthwise to expose the meat. In a small bowl using the back of a fork, mash the butter with the lemon juice, chives, and salt. Spread the butter over the lobster meat and between the shell and the meat. Grill the tails, shell side down, over direct medium heat, with the lid closed as much as possible, until opaque throughout, 8 to 10 minutes. Remove the meat from the shells and serve immediately with the warm sauce.
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