Mix basil, mayonnaise and butter in processor until basil is finely chopped and mixture is well blended. Season to taste with salt and pepper. (Can be made 1 day ahead. Cover and refrigerate.)
Cook bacon in heavy large skillet over medium-high heat until crisp, about 8 minutes.
Transfer to paper towels; drain.
Spread half of mayonnaise mixture over 1 side of 6 bread slices. Top each with 2 tomato slices.
Sprinkle tomatoes with salt and pepper. Top tomato slices with avocado, then with bacon strips, onion and lettuce.
Spread remaining mayonnaise mixture over remaining 6 bread slices.