Prepare the Salmoriglio:If you have a mortar and pestle, crush the garlic clove and basil leaves together until a paste is formed. Otherwise, finely mince the garlic clove and chiffonade the fresh basil. Place into a large bowl.
To the garlic and basil, add the lemon juice, zest, water, extra virgin olive oil, and capers. Whisk to combine, then season to taste with kosher salt and freshly cracked black pepper.
Remove the thawed lobster tail meat from the shells. Clean the tails of any vein or shell fragments.
Heat 1 tablespoon olive oil in a saute pan over medium high heat. Add the lobster tails to the pan and saute until cooked, about 2 minutes per side, flipping halfway through.
When the lobster tails have turned pink and opaque, remove them to a cutting board. Roughly chop the lobster into chunks.
Add the lobster pieces to the salmoriglio and toss to coat.
Split the ciabatta rolls in half, leaving the two halves connected on one side.Brush the inside of the rolls with basil oil (or plain olive oil).
Place the rolls, inside facing down, into the saute pan used for the lobster. Toast over medium-high heat, until the insides are golden, 1-2 minutes.(If you cannot fit all the rolls into the pan at once, work in batches).
Evenly divide the marinated lobster among the rolls. Drizzle with extra salmoriglio and sprinkle with chopped chives.Serve immediately.