Heat the oil in a medium frying pan over medium heat. Add the onion and garlic and cook, stirring frequently, for about 4 minutes or just until slightly softened, but not browned. Add the Veetee rice mix and cook, stirring, for 2 minutes or just until heated through. Remove from the heat and set aside.
Preheat and oil an outdoor grill or alternately preheat a stovetop grill pan.
Place the beans, basil, red bell pepper, chile flakes, and oregano in the bowl of a food processor fitted with the metal blade. Add the warm rice mixture along with the egg white. Season with salt and pepper and process, using quick on and off turns, until well-blended, but still a bit chunky.
Using your hands, form the mixture into 6 to 8 patties (depending upon the size you wish) of equal size. Brush each side with olive oil and place on the preheated grill (or in the grill pan). Grill, turning once, for about 6 minutes or until both sides are golden brown and crisp and the interior is hot.