Heat the oil in a skillet over medium heat. Add the mushrooms, cover and allow them to sweat over low heat until soft. Do not let them brown. Meanwhile, bring a large pot or skillet of water to a boil. Blanch asparagus by lowering it in a strainer into boiling water; cook just until bright green, about 30 seconds, and then transfer asparagus into ice water to stop the cooking. Set aside.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Grease a broiling pan.
To construct the rolls, lay the steaks out flat. If your steaks are thick, pound them to about 1/4 inch thickness. Brush soy sauce onto the surface, then place a few mushrooms, a couple of green onions and 3 asparagus spears at one end of each steak. Roll up towards the other end to enclose, and secure each bundle with a toothpick. Place the rolls seam side down on the broiling pan.
Roast under the preheated broiler until browned on top, about 3 minutes. Turn the rolls over and brown on the other side, 2 to 3 more minutes. Be careful not to overcook; this could burn the steaks or make the meat tough.
