Mix together beets, sirloin, salt, and pepper in a bowl. Form into 6 patties.
Coat burgers, cut sides of buns, and the grill rack (or nonstick skillet) with cooking spray. Over medium heat, cook burgers, turning, until golden brown, 4 minutes per side. In the last minute, top with cheese. Toast buns for 2 minutes.
Prepare eggs or pineapple as a topping. For eggs, heat 2 tsp. olive oil in nonstick skillet over medium heat; cook in batches until whites are set. Cook about 4 minutes for sunny-side-up eggs; for over-easy, flip eggs and cook 2 minutes more. For pineapple, spray with nonstick cooking spray and grill, 2 minutes per side.
Place burgers on buns, cheese side down, and top with egg or pineapple.
