Whisk together vinegar, olive oil, salt, and pepper in a large, shallow dish. Place mushrooms in dish, cover, and refrigerate for three hours, flipping mushrooms over halfway through.
Spray grill with olive oil and heat to 350 degrees. Place mushrooms on grill and grill until tender, about 8 minutes (depending on the type of grill you're using). Wipe grill clean.
Coarsely chop the artichoke hearts, tomatoes, roasted red pepper, and olives. Toss ingredients together in a small bowl.
Spread artichoke mixture onto 2 slices of bread. Place grilled mushrooms onto bread (you may have to slice them in half to get them to fit) and top mushrooms with arugula. Top with remaining bread slices.
Spray grill with olive oil. Place sandwiches onto grill or a panini press and spray tops with additional olive oil. Grill until golden brown, about 10-12 minutes. (If you don't have a grill that cooks from both sides, you'll have to flip the sandwiches after about 10 minutes.)
Notes
To grill the paninis, you can either use an actual grill, a panini press, or a grill pan on the stovetop.
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