Brown the Ground Beef
Heat a medium Dutch oven or heavy-bottomed pot over medium-high heat. Add the ground beef along with a generous pinch of salt. Spread the meat evenly and allow it to sear undisturbed for approximately 3 minutes. Break it into smaller pieces and continue cooking for another 3 minutes, stirring occasionally, until evenly browned.
Cook the Rice
Incorporate the basmati rice into the browned beef, stirring to coat each grain with the rendered fat. Continue cooking for roughly 3 minutes, stirring frequently, until the rice achieves a lightly golden hue.
Sauté the Onion
Add the chopped onion to the pot and cook, stirring often, for 5 minutes, or until the onion softens and turns translucent.
Season with Paprika and Garlic
Stir in the sweet paprika and minced garlic. Cook for 1 minute, allowing the spices to release their aroma.
Add Tomatoes and Sauerkraut
Pour in the canned whole tomatoes, crushing them gently as you add them. Introduce 6 cups of water, the sauerkraut, and the brine (or fresh cabbage with rice vinegar, if using). Add the parsley bunch and a pinch of salt. Bring the mixture to a boil, then cover loosely and reduce to a simmer for 15 minutes.
Final Adjustments
Taste and adjust seasoning with additional salt if needed. Remove and discard the parsley bunch. Serve the soup hot, garnished with freshly chopped parsley.