Heat the oil on a large griddle over medium-high heat and spread the bell pepper and onion in a thin layer.
Season the vegetables with 1/4 teaspoon salt and cook for 9 to 10 minutes until soft with dark brown edges.
Push the vegetables aside, add the shaved beef and season it with the remaining salt and smoked paprika.
Cook the beef for 3 to 4 minutes, turning and chopping it into short ribbons, until browned and at least 145°F.
Mix the vegetables through the meat, divide it into four piles and cover each with two cheddar slices.
When the cheese is fully softened, scrape each pile into a roll and press the sandwich closed for 30 seconds before serving.