Perhaps perfectly poached eggs and hollandaise might not seem exciting to some people anymore, but these eggs Benedict cannot be passed up. Why? Well, because there's lobster! (And some wonderful brioche doesn't hurt either.) Serve these with a side of roasted potatoes or vegetables.
For help with cooking the lobster, or anything lobster-related, click here to see The Ultimate Lobster Bible.
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*Note: If you don't have an egg-poaching pan, simply bring a medium-sized pot of water to a simmer and add a tablespoon or two of vinegar. Gently swirl the water with the back of a long spoon or spatula to create a whirlpool in the center, and drop an egg into the center of the whirlpool. With a slotted spoon, remove after about 3 minutes or when done to your liking. Repeat with the remaining eggs.
Preheat the broiler.
Remove the claws, knuckles, and tail from each lobster body and reserve the bodies for another use. Remove the meat from the claws, knuckles, and tails and reserve the shells. Cut the tail meat crosswise into ½-inch-thick slices; refrigerate along with the claw and knuckle meat until ready to use.
Melt 4 tablespoons of the butter in a sauté pan over medium heat. Add the shallots and cook until translucent, stirring occasionally, about 3 minutes. Add the reserved lobster shells and cook until the shallots are golden and any liquid has evaporated, stirring occasionally, about 5 minutes.
Off the heat, add the cognac, then return the pan to the stove and cook until the liquid has almost evaporated, about 3 minutes. Add the cream and cook until reduced by half and thickened, about 7-10 minutes. Meanwhile, set up a double boiler over very low heat. Strain the sauce through a fine-meshed sieve into the double boiler. Add the tarragon and lemon zest. Season with salt and pepper, to taste.
Lightly spread the cut sides of each brioche half with 1 teaspoon of the butter and broil until golden, 3-5 minutes. Turn off the broiler, cover the brioche halves with aluminum foil and keep warm in the oven.
Fill an egg-poaching pan with ½-inch water and bring to a simmer over medium heat. Lightly coat the poaching cups with nonstick cooking spray and crack an egg into each cup. Cover and cook until the whites are firm and the yolks are glazed over but still soft, about 3 minutes, or until done to your liking. Transfer the eggs to a plate and repeat with the remaining eggs.*
Add the reserved lobster meat to the sauce and heat until warmed through, about 3 minutes. Place 2 brioche halves on each of 4 plates. Using a slotted spoon, transfer the lobster meat to the brioche halves, dividing evenly. Top each with an egg, some of the sauce, and the chives. Serve immediately.