Spread some butter over 4 inches of 1 end of each breadstick. Spread Parmesan cheese on plate. Roll buttered end of each breadstick in cheese to coat lightly. Place 1 slice prosciutto on work surface; arrange 2 arugula leaves atop prosciutto, then place cheese-covered end of breadstick atop arugula and roll up prosciutto around breadstick, enclosing arugula. Repeat procedure with remaining breadsticks. Can be prepared 2 hours ahead. Place breadsticks in container, prosciutto ends up. Cover with plastic wrap and chill.