Preheat your grill to medium-high. Brush the steak with 1 Tbsp. of the oil. Grill steaks to desired doneness. Set the steak on a plate and cool to room temperature.
Place the remaining 1 Tbsp. of oil in a large skillet over medium-high heat. Add the mushrooms and sauté until tender and until any moisture in the pan has evaporated.
Mix in the chili powder, cumin, hot pepper sauce, salt and pepper. Cool the mushroom to room temperature.
Cut the steak, lengthwise, into 4 strips. Cut each strip, widthwise, into thin slices and place in a medium bowl. Toss in the mushrooms, bell pepper, cheese and cilantro or green onion. Place an equal amount of the steak mixture on one half of each tortilla. Fold each tortilla into a half-moon shape. Place a large skillet over medium, to medium-high heat. When hot, cook the quesadillas, in batches, 3 minutes per side or until golden and the cheese is melted. Cut them into wedges and, if desired, serve with salsa.