Prep the Veal: Pat dry the veal with paper towels. Season the veal cutlets with salt and pepper on both sides.
Preheat the Grill: Preheat your grill to medium heat.
Sauté the Onions: Heat the olive oil in a 12" skillet on the grill. Add the sliced red onions and sauté until soft, 8-10 minutes. Push them to one side of the pan.
Cook the Mushrooms: In the same skillet, add the butter to the cleared side of the pan. Add the sliced cremini mushrooms, minced garlic, thyme and stir. Cook until lightly browned, about 6-8 minutes.
Heat the Tomato Sauce: While the onions are cooking, warm the spicy tomato sauce in a small saucepan until heated through.
Toast the Buns: Toast the buns on the grill until golden and crisp. Remove from the grill and set aside.
Grill the Veal: Grill the veal cutlets for 1-2 minutes per side over medium heat, just until they are cooked through and have nice grill marks. Remove from the grill and let rest while you build the sandwiches.
Assemble the Sandwiches: Spread 3 tablespoons of tomato sauce on the bottom half of each bun. Sprinkle the Grana Padano cheese and grated mozzarella. Layer with 2 ½ pieces of grilled veal, mushrooms, and onions. Top with a handful of fresh arugula and the other half of the bun.
Serve: Serve the sandwiches with extra tomato sauce on the side for dipping.
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