Preheat the oven to 425 degrees F. Prepare a grill for medium-high heat. Brush the sweet potatoes with 1 tablespoon of the oil, and sprinkle with a large pinch of salt. Lightly oil the grill grates. Arrange the sweet potato rounds on the grill, cover with the lid and cook until there are nice grill marks and the potatoes are tender but not mushy, 8 to 12 minutes per side. Set aside to cool completely. (These "buns" can be refrigerated in an airtight container up to 2 days.)
Meanwhile, mash the beans and corn in a medium bowl until the beans form almost a paste. Add the cheese, 3 tablespoons of the cornmeal, the egg white, scallions and 3/4 teaspoon salt. Form the mixture into 4 patties about 1/2 inch thick. Refrigerate for at least 20 minutes up to overnight.
Heat the remaining 2 tablespoons oil in a large ovenproof nonstick skillet over medium heat. Put the remaining 2 tablespoons cornmeal onto a plate. Dredge both sides of the patties in the cornmeal. Cook until golden brown, 3 to 4 minutes per side. Transfer the skillet to the oven, and bake until the burgers reach an internal temperature of 165 degrees F, about 5 minutes.
Brush 4 of the grilled sweet potato slices with some of the mayonnaise. Put the lettuce leaves and burgers on the remaining 4 slices, and sandwich together.
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