Preheat oven to 425 degrees F. Heat the olive oil in a large, nonstick skillet over medium heat until hot. Add the mushrooms and cook, stirring, for five minutes until tender. Add the wine and cook 2-3 minutes or until the liquid is evaporated. Stir in the green onion, thyme, salt, and pepper. Remove skillet from heat and cool completely.
Heat the same skillet over medium heat until hot. Place the steaks in the skillet and cook for 3 minutes, flipping once (steaks will be partially cooked). Season with salt and pepper as desired.
Stack the phyllo sheets on a flat surface. Thoroughly spray each sheet with cooking spray. Cut the stacked sheets lengthwise in half, then crosswise in half to make 4 equal stacks. Place approximately 2 tablespoons of the mushroom mixture in the center of each phyllo stack, spreading the mixture to the diameter of the steaks. Place the steaks on top of the mushroom mixture. Bring all 4 corners of each phyllo stack together and twist tightly to close. Lightly spray each bundle with cooking spray and place on a greased baking sheet.
Immediately bake 9-10 minutes until golden brown, at 425 degrees F. Let stand five minutes. Serve immediately.