In a 6-quart Dutch oven or other heavy pot, combine beets, beans, and enough water to cover; season with salt and pepper. Bring to a boil; reduce to a simmer, and cook until beets are tender, 45 minutes to 1 hour (add more water to cover, if needed).
Transfer beets to a bowl; when cool enough to handle, peel and dice. Continue to cook beans until tender, about 1 hour more; pour off any excess liquid.
Finely chop garlic, and sprinkle with salt. Using a chefs knife, press a flat side of blade back and forth across garlic to make a paste. In a small skillet, heat oil over medium; add garlic paste, scallion, and cilantro. Cook, stirring frequently, until scallion has softened, 2 to 4 minutes.
Stir scallion mixture into beans; cook over medium until flavors have blended, about 5 minutes. Serve beans and beets over rice, topped with scallion and cheddar, with lime wedges alongside.
You can make this dish up to 3 days ahead; add the toppings just before serving.