Melt butter in a skillet over medium heat. Add mango and pineapple; sauté 5 minutes until fruit browns.
In a medium bowl, combine cooked rice, mango-pineapple mixture and salt; set aside.
In another medium bowl, combine 1 1/2 c. beans, garlic and cumin and mash with a fork.
Place 1/2 c. remaining beans and egg white in a food processor and process for about 20 seconds. Add bean purée, cheese and onion to mashed beans in bowl and mix well.
Shape bean mixture into a 1/2-in. thick patty. Pour cornmeal in a shallow dish and cover both sides of patties with cornmeal.
Heat olive oil in a fry pan over medium heat. When water sprinkled into the pan sizzles, start cooking patties, two at a time, about 2 minutes on each side or until golden brown.
To serve, spoon about 1/2 c. rice onto each of 4 plates; top each serving with 1 patty and 1 Tbsp. sour cream.