I thought that borscht was uniquely Russian and used beets.
My Russian-style borsch goes like this:
throw meat or bones into water with some salt and boil a while
add potatoes and thinly sliced cabbage (unless you want to use sauerkraut instead)
when the potatoes are cooked, grate carrots and beets, and chop onion
fry the carrots and beets and onion a little in a frying pan to brown them slightly and give them a caramelized flavor
add them to the pot, along with a little tomato paste and some vinegar (leave out the vinegar if you chose to use sauerkraut instead)
(if you chose to use sauerkraut instead, now is when you add it.)
then you cook until everything is thoroughly red
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As I understand it, the common thing that is eaten in the Ukraine is called Green Borsch. It has meat and potato and diced hardboiled egg and sorrel.
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I spell it "borsch" because in Russian that is how it sounds. No "T". Russian has 2 sounds that we would transcribe in English as "sh". They refer to one as hard and the other as soft. "Borsch" has the hard one, so we generally transcribe that as "sch".