Light your grill and set it up for two-zone cooking.
Slice each lobster tail in half, leaving the very end uncut. Open the tails like a book so the tail meat from both sides is facing up. Season with salt and set aside.
Slice all of your butter into small cubes. Place a medium saucepan on medium-high heat. Add one tablespoon of water. When it simmers, reduce the heat to medium and add 3 to 4 cubes of butter. When the butter is almost melted, add 3 to 4 more cubes until it is all gone. The sauce should never simmer (if it gets too hot add more butter to cool it down and remove it from the heat). Once all of the butter is melted, remove from the heat and add your tarragon, two tablespoons of the chives, and salt to taste. Give it a good stir and put it in a thermos to keep it warm without breaking.
Start the lobster with meat face down on the direct heat portion of the grill. Leave them for 2 to 3 minutes, or until they have a bit of char, and then flip them over and move them to the indirect side of the grill. Once the internal temp hits 140 degrees F remove them from the grill.
Remove the lobster meat from the shell and set the shells aside. Cut the meat into bite-size pieces and place it in a mixing bowl. Add enough of the butter sauce to coat. Give it all a good stir and then spoon it back into the tail shells.
Top with a squeeze of lemon, extra chives, and a pinch of salt.