Using a food processor, grind the giardiniera, arugula and olives.
Coat 1 side of the bread slices with the butter. Build each sandwich as follows: a bread slice (buttered side out), giardiniera relish, a cheese slice, 2 prosciutto slices, 2 soppressata slices, 2 salami slices, another cheese slice, more relish and another bread slice (buttered side out).
Preheat a griddle or cast-iron skillet over medium heat. Add the sammies and cook, turning once, until the cheese melts, 6 to 8 minutes.