Kill the lobsters with a sharp-pointed knife plunged into the body behind the head.
Remove the bands from the claws. Twist off the tails, and carefully split lengthwise. Twist the claws and knuckles off the bodies; separate the claws from the knuckles. Crack the claws a little using a sharp blow with the back of a knife. Split the bodies. Remove the intestinal tract from the tail meats. Arrange the lobsters in a single layer in an ovenproof baking dish, tail and body shells down. Sprinkle sea salt (preferably from Maine) and black pepper over the lobster meat, followed by a generous dressing with the olive oil.
Preheat the oven to its hottest temperature setting. (See Cook's Note)
In a non-reacting skillet filmed with a little melted butter, saute the mushrooms over high heat for 2 or 3 minutes. Season with salt and pepper and set aside.
Place the lobsters in the oven, and roast until the shells are almost uniformly red and the meat is opaque, as long as 7 or 8 minutes in a home oven.
Remove the lobsters from the oven. Add the mushrooms and shallots. Pour the hard cider over the lobsters, and scatter the butter pieces around the dish. Place the lobsters back in the oven, and continue roasting, checking frequently, until the butter has completely melted, about 2 or 3 minutes. Swirl the pan to combine the liquid ingredients into a saucy emulsion. Check the seasonings and adjust with more sea salt and pepper if necessary. Serve the lobsters on dinner plates and spoon the sauce over them.
Our wood-burning oven achieves intensities of over 800 degrees F, but most home ovens don't exceed 500 degrees F.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.