Heat oil and butter in a large skillet over medium-high heat. Saute onion, chile, garlic, and 1/4 teaspoon salt until onions are translucent, 2 to 4 minutes. Add mushrooms and cook, stirring, over high heat, until mushrooms turn deep golden brown and release their liquid, 4 to 6 minutes. Remove from heat and sprinkle with oregano. Adjust salt if necessary.
Toast tortillas over the flame of a gas burner, turning with tongs, until crisp and blackened in spots. (You can also char them in the broiler.) Divide mushroom mixture among tortillas and sprinkle with Parmesan.