Mix basil, mayonnaise and butter in processor until basil is finely chopped and mixture is well blended. (see note in summary)
Season to taste with salt and pepper.
(Can be made 1 day ahead. Cover and refrigerate.)
NOTE: as a recipe for two we used 1/4 cup pre chopped basil to 1/2 cup mayo and 1 1/2 Tbs. butter. Had extra left over but taste was very good.
TIP: Make the mayo a few hours or more before making the sandwich to allow the mixture to thicken.
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.
Place bread slices atop lettuce.
Cut sandwiches in half and serve.