Place mayonnaise and lemon juice in a large bowl. Season with salt and pepper and stir together until smooth.
In another mixing bowl, add half of the chickpeas and mash with a potato masher or the back of a fork until coarsely mashed.
Add mashed chickpeas, whole chickpeas, shallots, carrot, and green onion to mayonnaise mixture and fold together until fully mixed together. Season with salt and pepper.
Place a couple leaves of green leaf lettuce atop two slices of bread (if using) and top with chickpea mixture. Top each chickpea salad mixture with remaining bread and gently press down. Slice each sandwich in half and serve.