Poaching the lobsters can be done ahead of time. Combine water, vinegar and salt and bring to a boil. Cook each lobster separately. Plunge the lobster into the boiling water and cook for two minutes. Do not overcook since the meat should be rare. When the lobster is done, immerse in an ice bath and allow to rest. Remove the meat from the tail and claws and refrigerate until ready to use.
Heat the beet juice over medium heat in a small saucepan and reduce it by half. Add the balsamic vinegar, lemon juice and horseradish; set aside.
Cut one pound of butter into small cubes and let it come to room temperature. Heat one tablespoon of water in a small saucepan over low to medium heat; whisk in the butter cubes, one by one, until the sauce is completely emulsified. When this is done, put aside and keep it in a warm place. Overheating will cause it to separate. This will be the base for the poaching liquid for the lobster.
Place the beurre monte over medium heat. Make sure it does not boil, as it will separate. Put the rare lobster meat into the butter and leave for 5 to 6 minutes. Reheat the beet sauce. Bring the beet sauce to a simmer and season.
Add a bit of beurre monte gradually, stirring until the beet sauce coats the back of a metal spoon. Remove the lobster from the butter with a slotted spoon. Slice half a tail for each person. Put some of the beet sauce on the bottom of each plate along with a piece of lobster. You can garnish with a touch of extra virgin olive oil around the rim of the plate and a few salad greens on top of lobster.