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1
Combine olives, capers, peppers, parsley, giardiniera, and garlic in the bowl of a food processor. Pulse to chop until no pieces larger than 1/2-inch remain. Transfer to a bowl. Add vineger and olive oil and stir to combine. For best results, let olive salad rest overnight before continuing to step 2.
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2
Melt one third of butter in a large non-stick skillet over medium heat until foaming subsides. Add both bread slices and cook, swirling occasionally, until pale golden brown on bottom side, about 2 minutes.
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3
Transfer to a cutting board toasted-side-up. Place one cheese slice on top of one slice of bread. Spread a layer of olive salad across bread with a spoon, then top with soppressata, mortadella, and capicola, followed by second cheese slice. Close sandwich, with both toasted sides facing inwards.
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4
Melt one more piece of butter in the skillet and reduce heat to medium low. Add sandwich and cook, swirling occasionally, until deep, even golden brown, about 5 minutes. Remove sandwich using a flexible metal spatula. Add the remaining butter. Return sandwich to skillet cooked-side up. Season with salt. Cook, swirling occasionally, until second side is deep, even golden brown and cheese is thoroughly melted, about 5 minutes. Serve immediately.
