In a large (12-inch) nonstick skillet, heat the oil over medium-high heat. Add the pork and cook, turning occasionally, until browned, about 6 minutes. Transfer to a plate.
Add the onion, pepper, and mushrooms to the skillet. Cook, stirring often, until the mushrooms give off their liquid and it almost completely evaporates, about 6 minutes. Add the garlic and cook for 1 minute.
Sprinkle the vegetables with the flour, paprika, caraway seed, and marjoram and reduce the heat to medium. Stir until all of the ingredients are well coated, about 1 minute. Stir in the broth and bring to a simmer. Return the pork to the skillet. Reduce the heat to low and simmer, uncovered, until the pork is tender, about 20 minutes. Season with salt and pepper to taste.
Spoon the rice onto individual dinner plates, top with the goulash, and add a dollop of sour cream on each serving. Serve immediately.