Heat a grill to medium-high. Clean and lightly oil hot grill. Cut onion into thin rings and season with salt and pepper. Grill until lightly charred and crisp-tender, about 4 minutes per side. Transfer to a plate. Form ground beef into 4 patties and season with salt and pepper. Grill 4 minutes per side for medium. Transfer to bottom halves of buns. Top each burger with an egg, grilled onions, your choice of sauce, sliced beets, and lettuce.