Preheat the oven to 350°F (175°C) and lightly spray a mini muffin pan with nonstick cooking spray.
Unroll the crescent dough sheet and press it into one large rectangle, sealing any seams; cut the rectangle into 24 equal squares.
Press each dough square into a muffin cup and shape the edges upward to form a small cup.
In a mixing bowl combine the chopped roast beef, honey mustard, hot sauce, teriyaki sauce, shredded provolone, and the egg; mix until evenly combined.
Spoon the meat-and-cheese mixture into each prepared dough cup, filling evenly.
Bake for 15–17 minutes, or until the dough is lightly browned and the filling is set; remove from oven and let cool slightly before serving.