Heat oven to 400 degrees Fahrenheit.
In a small bowl, combine basil, shallot, melted butter, salt and pepper.
Brush mixture on lobster tails and coat with toasted panko crumbs.
Heat in oven six to eight minutes, then broil for 30 seconds.
To a medium saute pan over high heat, add 2 tablespoons butter, 2 tablespoons olive oil, tomatoes, chives, parsley, lemon juice, 1/2 tablespoon vinegar, salt and pepper.
Reduce by one-quarter and add lobster.
Cover meat completely with vinaigrette and season again.
In a large bowl, combine pea shoots with 1/2 tablespoon olive oil, remaining vinegar, salt and pepper.
Reserving vinaigrette, place half a lobster tail in each of 6 bowls; add equal amounts of pea-shoot mix.
Add rest of lobster meat, divided equally among six portions, and about a tablespoon of warm tomato vinaigrette per serving.
Drizzle with olive oil.