Preheat barbecue to medium-high. Set a cast iron (or heavy) skillet on one side of barbecue. Season steak with 1/4 tsp (1 mL) each salt and pepper. On other side of barbecue, grill steak for 3 to 5 min. per side while basting with barbecue sauce. Remove steak and let rest.
Whisk together eggs and milk with remaining salt and pepper in a medium bowl. Add butter to hot skillet, pour in egg mixture and stir as it cooks. When eggs are almost set, sprinkle with green onions and finish cooking. Remove from heat and set aside.
Toast buns lightly on barbecue. Thinly slice steak, divide it and eggs among buns and serve.
The sliders are perfect for using up leftover steak.