Combine all of the ingredients for the barbecue sauce in small pot and bring it to a boil over medium heat. Reduce the heat to low and cook until thickened, about 30 minutes. Cool completely and store in the refrigerator for a make-ahead meal.
Reheat the sauce over medium-low heat to warm.
Heat the stock in a skillet over medium heat until boiling. Reduce the heat to low, add the roast beef and simmer until the meat is warmed through but not dried out. When the liquids have just about evaporated, pour the barbecue sauce over the meat and stir gently to combine.
Lightly toast or warm the bread. Pile the barbecue meat into the sub rolls and top with pickled chopped vegetables.
Add the arugula, the cheese, tomatoes and basil to medium salad bowl. Douse with extra-virgin olive oil and season with salt and pepper, to taste. Toss to combine and serve with the beef sandwiches.