For stuffing, in a large skillet melt butter. Add mushrooms, onion, and thyme; cook until onion is tender. Remove from heat; stir in crushed croutons, prosciutto, cheese, and the 1 tablespoon balsamic vinegar. Season with pepper. Meanwhile, in a small saucepan, bring remaining balsamic vinegar to boiling. Boil, uncovered, until reduced by half, about 6 minutes; remove from heat.
Prepare medium-hot banked fire in covered grill or preheat gas to medium-high. Lightly oil grill grate.
To butterfly the pork loin, place fat-side down. Make a single lengthwise cut down the center of the pork, cutting to within 1/2 inch of the bottom side. Spread open. Make another lengthwise slit to the right of the first cut and a third slit to the left. Cover pork with clear plastic wrap. Pound with a meat mallet to even thickness, about 3/4 inch. Remove plastic wrap.
Spread stuffing onto pork, leaving a 3/4-inch border. Starting from a long side, roll the pork and stuffing jelly-roll style. Tightly and evenly retie meat with butcher's twine or cotton kitchen string at 2-inch intervals. Sprinkle with salt and pepper.
Grill pork, fat-side up, over indirect heat, away from the fire until internal temperature on a thermometer reads between 145 degrees F. (medium rare) and 160 degrees F. (medium) – start checking the temperature at 45-50 minutes – brushing with reduced balsamic twice during last 10 minutes of grilling. Remove pork from heat; let rest about 10 minutes. To serve, remove twine and cut pork into slices, about 3/4-inch thick.
Serves 8 to 10 servings
To roast in lieu of grilling: Prepare and stuff pork loin as directed. Preheat oven to 325 degrees F. Place stuffed roast, fat-side up, on a rack in a shallow roasting pan. Roast, uncovered, until internal temperature on a thermometer reads between 145 degrees F. (medium rare) and 160 degrees F. (medium), about 50-60 minutes, brushing with reduced balsamic twice during last 10 minutes of roasting. Let rest about 10 minutes and slice as directed above.