Prepare the grill for direct cooking over medium heat (350° to 450°F).
Spread the bacon out in a 12-inch cast-iron skillet and fry over direct medium heat, with the lid closed, until crisp, 15 to 17 minutes, stirring the bacon pieces occasionally. Using a slotted spoon, transfer the bacon to paper towels to drain. Carefully remove about 2 tablespoons of the bacon grease from the skillet and discard.
Combine the onion, beets, salt, pepper, and thyme in the skillet and cook over direct medium heat, with the lid closed, for 10 minutes, stirring once. Add the garlic and potatoes and cook, with the lid closed, until the beets and potatoes are tender and there are browned bits sticking to the skillet, 25 to 30 minutes, stirring and scraping the bottom of the skillet occasionally.
Stir in the vinegar and all but ¼ cup of the bacon pieces. Top the hash with the cheese and the remaining ¼ cup of bacon. Serve immediately with scrambled or fried eggs, if desired.