Preheat the oven to 350 degrees F.
Season the entire beef top round with the barbecue seasoning and set aside. Heat the olive oil in a large saute pan over high heat. Sear the beef on all sides until browned. Transfer the beef to a roasting pan and finish in the oven until cooked to medium, with an internal temperature of 130 degrees F, about 10 to 15 minutes. Allow the beef to rest before slicing. While the beef cooks, heat the beef broth in a saucepan with the thyme sprigs. Simmer over medium heat until reduced by half and then turn down the heat and keep warm. When the meat is cooked and rested, slice the beef thinly and add it to the pan of broth. Coat the beef in the broth. Warm the tortillas in a pan or in the oven. Top each tortilla with a few spoonfuls of cheese sauce, lettuce, then a few slices of beef and then some of the Giardiniera, and peppers. Roll up into a sandwich wrap and enjoy!
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.