In a small bowl, whisk together the vinegar, 3/4 teaspoon salt, garlic, rosemary, and pepper. Gradually whisk in 1/2 cup of the oil to make a dressing. Set aside.
Preheat a stove top grill pan or out-door grill to medium heat. Brush the peppers, zucchini, eggplant, mushrooms, and onion with the remaining oil. Grill the peppers until lightly browned and tender, about 5 minutes per side. Transfer the peppers to a pan or large platter. Grill the zucchini, until softened and lightly browned and tender, about 3 minutes per side; transfer to the pan with the peppers. Grill the mushrooms, until soft but not flabby, about 2 minutes per side; set aside with the other vegetables. Grill the onions until soft, about 5 minutes per side; set aside with the other vegetables. Drizzle a tablespoon of the dressing over the vegetables and season with salt and pepper to taste.
To make the Dagwood sandwich: Split the bread in half to make two evenly thick square pieces. Brush the dressing on the cut sides. Make a layer of each of the grilled vegetables that evenly covers one piece of the bread, overlapping the pieces slightly, if necessary. Top with a layer of the tomatoes, season with salt and pepper. Lay the arugula on top and drizzle them with about a tablespoon of the dressing. Finish the sandwich off with the cheese and lay the other piece of bread on top. Wrap the sandwich tightly in plastic wrap. Place the wrapped sandwich on a cutting board or baking sheet and weigh it down with a heavy object, such as a cast-iron skillet, for about 1 hour. When ready to serve cut into 4 to 6 pieces equal pieces.
This is a great way to use up a variety of leftover grilled vegetables. In keeping with a true Dagwood style, feel free to improvise with the vegetables and cheese.
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