Combine the champagne, lime juice, vanilla bean (including the pasty insides and the pod), shallots, and sugar in a medium skillet over high heat. Bring to a boil, breaking up the vanilla bean pasta with a whisk to disperse it into the liquid, and simmer until the liquid is reduced by about half.
Stir in the cream, salt, and pepper and cook, stirring occasionally, until the sauce is thick enough to coat a spoon, for about 5 minutes. Whisk in 1 tablespoon of the butter, remove the skillet from the heat, and whisk until the butter is thoroughly incorporated. Strain through a sieve or cheesecloth, and keep warm until ready to serve. Makes 1 cup.
Heat the remaining 2 tablespoons butter in a large skillet over high heat. When the butter is half melted, add the lobster and sauté, tossing occasionally until the lobster meat is just warmed through, for about 1 minute. Remove the skillet from the heat, add the chives, and toss to mix well.
To serve, pour 1/4 cup of the sauce into each of 4 dinner plates. Divide the mushroom salad into 4 equal portions and heap it in the center of each plate. Divide the lobster meat into 4 equal portions and arrange it around the mushroom salad.
Makes 4 main-course servings.