Select a large soup pot and heat the water to boiling over high heat. Add 2 teaspoons of salt and the potatoes and reduce to a simmer.
Meanwhile, warm half the olive oil in a saute pan over medium-high heat. Add the onion and cook, stirring, until golden. Add the pepper halves for the final shake, then transfer all to the simmering potato pot. Repeat with separate batches of carrots and then beets, adding more oil as necessary in between and sauteing each vegetable until a bit softened and a shade deeper in color (about 3 minutes), before adding to the soup.
Keep the pot simmering until all the vegetables are fully soft and beginning to merge, about 20 minutes more. Season with lemon juice, salt, pepper, and dill to taste. (The seasoning brings out much of the soup’s flavor, but start small and continue tasting until the balance is just right.)
Remove soup from heat and let cool. Borscht is best after it has chilled for several hours and is then reheated, but still tastes great directly after cooking.
Offer with sour cream and additional minced dill.