To make the olive salad, combine all the ingredients in a bowl. If possible, let the mixture sit for at least 2 hours so the flavors can meld together.
To make the panini, preheat a panini press on medium high heat.
Spread the outer edges of the Italian sandwich bread with the softened butter, then spread a spoonful of the olive salad on the inside of each piece of bread. Place two slices of cheese on top of the olive salad on each piece of bread. Then layer on a couple slices of mortadella, genoa salami, and deli ham. Close the sandwich and place on the panini press.
Press the lid of the panini maker down firmly, and cook the sandwich for 3-4 minutes until golden brown and crisp. Serve and enjoy!