Preheat fan oven to 200 degrees Celsius. Place oven rack at the highest point possible in the oven for you to bake your lobsters.
Place butter in a small saucepan and gently sauté the shallot until soft. Add wine and cook until almost all the liquid has evaporated. Add cream stirring well to combine.
Next add cayenne pepper, mustard, tarragon, chives. When combined add Gruyère cheese, allow this to melt then remove sauce from heat.
Keeping the lobster meat inside its shell use a small sharp knife gently cut the lobster meat at intervals along the tail. This will allow some sauce to fall through the meat. Place lobster halves in a small baking tray making sure the lobster halves stay upright.
Spoon 1/2 of the sauce over each lobster half.
Bake for 10-15 minutes or until the top is golden and the lobster meat is cooked through.
Serve with golden roasted potatoes and fresh tarragon.