Heat the vegetable oil in a large heavy-bottomed pot or Dutch oven over medium-high heat. Add the pork ribs and sear them for 5 to 7 minutes until they are deeply browned on all sides. Remove the ribs to a plate and set aside, leaving the rendered fat in the pot.

Lower the heat to medium and add the chopped onion to the pot. Sauté for 4 to 5 minutes until softened and translucent, using a wooden spoon to scrape up any browned bits from the bottom. Add the minced garlic and cook for 30 seconds until fragrant.

Stir in the grated beets and carrots, cooking for another 5 minutes until the vegetables begin to soften. Add the tomato paste and stir well, cooking for 2 minutes to slightly caramelize the paste and deepen its flavor.

Return the seared pork ribs to the pot and pour in the beef broth. Bring the mixture to a gentle boil, then reduce the heat to low, cover, and simmer for 45 minutes to tenderize the meat and build the broth's savory foundation.

Add the cubed potatoes and shredded cabbage to the simmering soup. Cover and continue to cook for 25 to 30 minutes, or until the potatoes are fork-tender and the pork is practically falling off the bone.

Stir in the white vinegar to brighten the soup and balance the sweetness of the beets. Season with the salt and black pepper, tasting and adjusting as necessary, then simmer for a final 5 minutes to let all the flavors meld together.

Ladle the hot bortsch into deep bowls, ensuring each portion gets a generous, tender pork rib. Serve hot, garnished with a dollop of sour cream and a sprinkle of fresh chopped dill, accompanied by thick slices of crusty rye bread.