Heat 1 Tbsp olive oil over medium-high heat, and cook sliced sausage rounds for a few minutes on both sides. Transfer to a plate.
Heat the remaining 2 Tbsp olive oil on medium-high, and add the chopped onion, garlic, and carrot. Season with salt, pepper, and oregano. Cook until vegetables are tender, and stir occasionally.
Add chicken stock to the skillet, rice and chopped tomato. Cook rice mixture according to package instructions for the rice. Bring to a boil, stir again, then cover and simmer on low heat until liquid has evaporated and rice is cooked.
Add sauerkraut to the skillet and stir. Serve warm.