Pat pork dry and season with salt and pepper. Heat oil in a large skillet over medium-high heat and brown pork in 2 batches for about 6 minutes each, transferring it to a 6-quart slow cooker.
Cook onion in the skillet for 4 minutes. Lower the heat and stir in garlic, paprika and caraway for 30 seconds, then immediately add broth and scrape up the browned bits.
Pour the skillet mixture over the pork and add peppers and tomatoes. Cover and cook on low for 7 to 8 hours until pork reaches at least 145°F and is fully fork-tender, usually at a much higher temperature.
Mix cornstarch with cold water and stir into the stew. Cover and cook on high for 20 to 25 minutes until lightly thickened.
Switch off the heat. Mix sour cream with a little hot stew liquid in a small bowl, then fold it back into the pot and divide among 6 bowls.