In a medium bowl, coarsely mash beans with a fork and add flour until well combined. Shape bean mixture into 4 patties. Melt butter and oil together in a large skillet over medium heat. Add bean patties, in batches as necessary, and brown on both sides until crispy.
Spread mayonnaise on 2 pieces of white bread. Place bean patty on bread and top with onions. Repeat to make 4 sandwiches.