Place chives, mayonnaise, tarragon, and yogurt in a blender. Add zest and juice of 1 lemon half, then add 2 tablespoons of extra-virgin olive oil; season with salt and pepper. Purée until smooth.
Squeeze juice from remaining lemon half over lettuce and cucumbers in a medium bowl. Drizzle with oil; season with salt and pepper. Toss to coat.
Divide and smash avocado between 2 slices of bread; season with salt and pepper. Arrange lettuce and cucumbers over. Top with mozzarella and sprouts. Spread remaining 2 slices of bread with ¼ cup green goddess dressing and close sandwiches.