In a medium bowl, combine the beans, oil, salt, and pepper. Roughly mash the mixture with the back of a fork.
Place 8 of the bread slices on a work surface. Divide the mashed beans among the slices. Top with the onion, cucumber, sprouts, and avocado. Stack the open-faced sandwiches on top of one another, avocado-side up, to make 4 double-decker sandwiches. Top with the remaining 4 slices of bread. Slice each sandwich in half, if desired, and serve.
Tip: You can make this quick sandwich even more quickly by using store-bought hummus in place of the smashed white beans.