Pour oil into medium nonstick skillet, add butter.
Place over medium heat; when butter is melted, add prosciutto; cook 1 to 2 minutes, until prosciutto is fairly crisp; remove from heat immediately; remove prosciutto from skillet to small plate or bowl, leaving oil in skillet.
Break or cut meat into small pieces (I use kitchen shears) & set aside.
Return skillet to medium heat; add asparagus and mushrooms; sauté 8 (or so) minutes, adding salt and pepper toward end of cooking time (go easy on the salt because the prosciutto and cheese are both salty!) Stir in crispy prosciutto bits.
Remove to warm serving dish; sprinkle with cheese. Serve immediately.
Recipe may be multiplied. Cover and refrigerate leftovers.