Puree avocado, parsley, tarragon, lemon zest and juice, oil, garlic, anchovies, and 1/4 cup chives in a blender. Season with salt and pepper. Thin with additional lemon juice or water if needed. Cut remaining chives into 1/2-inch pieces. Toss together lettuce and half of the dressing; arrange on a platter. Top with cucumbers, tomatoes, and remaining chives. Drizzle with remaining dressing. Season with pepper.