Mix the ketchup, mustard, cherry pepper, and vinegar in a small bowl and set aside. In a food processor, pulse the beef until just coarsely chopped. Transfer to a bowl. Toss with 1/2 tsp. salt and 1/2 tsp. 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. Transfer to a plate. Heat the remaining 1-1/2 Tbs. oil in the pan. Add the onion and 1/4 tsp. salt; cook, stirring often, until beginning to soften and brown, about 4 minutes. Add the mushrooms and 1/4 tsp. 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. Split Turano French Rolls almost through lengthwise and pile on the steak. Top with cheddar cheese.