Warm 1 tbsp. olive oil in a large skillet over medium heat. Sauté onions and salt until they begin to soften, about 3-4 minutes. Add remaining tbsp. olive oil, mushrooms and garlic and continue to cook until mushrooms are tender, about 10 minutes more. Stir in green chiles and cook an additional 2 minutes. Remove pan from heat.
Lay out tortillas on a clean work surface and layer each with spinach, mushroom mixture and cheese. Fold in one end then roll tightly to create wrap.