Preheat oven to 175°C (350°F) and lightly grease a 23x33 cm baking dish.
Heat vegetable oil in a large skillet over medium-high heat; add ground beef and cook until browned, breaking into small crumbles.
Add diced onion and garlic to the beef; sauté for 5 minutes until onions are translucent.
Stir in tomato paste, paprika, salt, and black pepper; cook for an additional 2 minutes then remove from heat.
Spread half of the drained sauerkraut in an even layer at the bottom of the prepared baking dish.
Layer the cooked rice over the sauerkraut, followed by the seasoned beef mixture.
Distribute the remaining sauerkraut over the beef layer.
Spread the sour cream evenly across the top of the sauerkraut layer using a spatula.
Sprinkle the shredded cheese uniformly over the sour cream layer.
Bake for 35 to 40 minutes until the casserole is heated through and the cheese is golden and bubbling.