Use a pair of sharp kitchen shears to cut down the middle of the lobster shell. Pry the sides of the shell away from the lobster meat inside. Loosen and then remove the lobster tail meat. Remove the vein along the top of the lobster if it's still intact. Set the meat aside. Use the shears to cut the tail shells into several smaller pieces.
In a saucepan, heat the olive oil over medium high heat. Add the shallot and sauté for 3 to 4 minutes until softened. Add the minced garlic and cook briefly. Then add the lobster shell pieces. Cook the shells until they begin to change colour, another 2 to 3 minutes.
Deglaze the pan with the white wine. Add the fresh tarragon sprigs. Bring to a boil and allow the wine to reduce to 1/2 cup. Strain the wine and return to a clean saucepan.
Cut the butter into small pieces. Bring the reduced wine to a gentle simmer and then turn down the heat so it isn't bubbling. Begin to whisk the cold butter into the reduction. Whisk constantly as you add the butter a few pieces at a time. Do not allow the liquid to come to a boil or the sauce may split. Whisk in the lemon juice, kosher salt, and pepper.
Be sure to keep the beurre monté at a temperature below 180° F. Do not allow it to boil. If the sauce begins to split, reduce the heat and whisk in a few more pieces of cold butter – that should emulsify the sauce once more.
Transfer the lobster tails to the warm beurre monté. Gently poach the lobster for 10 to 12 minutes until the lobster is cooked, flipping them several times as needed. Remove the lobster to a platter and cover to keep warm.
Toast the bread and set aside. Once the lobster is removed from the beurre monté, whisk in the chopped tarragon and chives. To serve, slice the lobster tails in half lengthwise. Arrange two halves on a slice of toast and spoon over some of the beurre monté. Garnish with more chopped fresh chives. Enjoy!