Mix together the soppressata and cheese in a medium bowl. Lay out 4 slices of the bread on a work surface and divide the meat and cheese mixture evenly among the bread, about 1/3 cup each. Close the sandwiches with the remaining bread. Thinly and evenly spread the butter on both sides of the bread using about 1 1/2 tablespoons of the butter for each sandwich.
Place 2 sandwiches into a medium skillet over medium heat. Cover the skillet with a lid and cook until the bread is evenly golden brown, about 2 minutes. Flip, cover again and cook until the bread is golden brown and the cheese is visibly melted, about 2 minutes more. Repeat with the remaining sandwiches. Serve with fig jam on the side.