Preheat a grill to medium high. Make the green goddess sauce: Combine the mayonnaise, chives, parsley, tarragon, capers, anchovy, garlic, lemon juice and a pinch each of salt and pepper in a small food processor. Puree until bright green and smooth. Transfer to a small bowl and stir in the sour cream. Season with salt and pepper. Thin with water if needed. Refrigerate until ready to use.
Make the sandwiches: Season the chicken with salt and pepper and rub with the olive oil. Grill, flipping once, until well marked on both sides and cooked through, 6 to 8 minutes. Remove to a plate. Grill the buns until toasted.
Serve the chicken on the buns with the green goddess sauce, tomatoes, bacon and lettuce.