Mix the ketchup, mustard, cherry pepper and vinegar in a small bowl and set aside.
In a food processor, pulse half of the beef until just coarsely chopped. Don't skip this quick-chop step. Transfer to a bowl. Repeat with the remaining beef. Toss the beef with 1/2 teaspoons salt and 1/2 teaspoons pepper.
Heat 1-1/2 Tbs. of the oil in a heavy 12-inch skillet over medium-high heat until shimmering hot. Add the beef and cook, stirring occasionally, until it loses most of its raw color, about 3 minutes. Careful not to overcook. Transfer to a plate. Heat the remaining 1-1/2 Tbs. oil in the pan. Add the onion and 1/4 teaspoons salt; cook, stirring often, until beginning to soften and brown, about 4 minutes. Add the mushrooms and 1/4 teaspoons salt; cook, stirring often, until they soften and begin to release their juice, about 2 minutes. Add the ketchup mixture and beef and any accumulated juices. Cook, stirring often, until heated through, about 2 minutes. Stir in the cheese until melted, about 1 minute more.
Split the rolls almost through lengthwise and pile on the steak and arugula. Cut the sandwiches in half and serve.