Line baking sheet with parchment paper.

Purée beans, rice, egg whites, paprika, salt, pepper, and cayenne in food processor till smooth. Put mixture in large bowl.

Add sweet potato, corn, reserved black beans, and cornmeal to bowl; combine completely.

Scoop mixture onto the parchment paper (1/2 cup each) shape so 1/2-in thick.

Heat oil in a large nonstick skillet over medium. Sauté burgers about 5 min each side.

Assemble burgers on toasted buns with cilantro mayo, lettuce, onion, and fried eggs.

Cook sweet potato fries.

Pulse together mayo, cilantro, garlic, and lime juice in food processor; season with salt, pepper, and cayenne.