Cut the beef chuck into bite-sized pieces and place it into a small pot with just enough water to cover the beef by about an inch.You can also use a bone in beef shank instead of the chuck.
Bring the water to a boil, reduce heat to low and cook at a simmer for 1 - 1 ½ hours, until fork tender, while cooking the Borsch. You can also add some black peppercorns and bay leaves to flavor the broth. Skim off the foam from the top of the beef broth as it is cooking.
While the beef is cooking, prepare the rest of the Borsch in a large pot or Dutch oven.
Heat the butter or oil and add the onion, carrot, and celery. Season with salt and ground black pepper. Cook on medium-low heat for 6-8 minutes.
Pour in the water and chicken broth. Add the potatoes and cabbage to the Borsch. Bring to a boil, season with salt and keep cooking, at a simmer until the potatoes are cooked through, 15-20 minutes.
Meanwhile, place the beets in a large skillet with the tomato sauce, 1 cup water, lemon juice and sugar. Season with salt and ground black pepper. The lemon juice will keep the beets a beautiful, vibrant red and also give it a slight tang to the Borsch. The sugar balances out the acidity of the lemon juice.
I prefer using roasted beets for Borsch, but you can also cook the beets in a pot with water or even use raw beets, but you will have to cook them much longer on the skillet. If you're sautéing raw beets, you will need to add more water to the skillet and cook the beets longer until they are completely cooked through and tender.
Cover the skillet, bring the water to a boil, then reduce the heat and simmer on medium heat for about 15 minutes, until most of the water and tomato sauce is absorbed into the beets.
Clear a space in the center of the skillet and add the minced garlic to the beets, cooking for another minute or two, just until the garlic is cooked through.
When the potatoes are tender, add the beets to the soup and take Borsch off the heat. By this time, the beef should be fork-tender.
If you like the consistency of the soup, drain the beef, if you like the soup to be thinner, drain the beef through a fine mesh sieve and add the beef liquid to the soup as well. If you used beef shank, remove the meat from the bone and cut it into bite-sized pieces. Add the beef to the Borsch as well.
Season with salt and pepper, if needed. Garnish with finely chopped fresh dill and green onions. We Russians like to add a dollop of sour cream to our bowl of Borsch.