Preheat the grill and brush with a grill brush to remove any excess crud and fat. After brushing, run an oiled towel over the grates to pick up any loosened crud or soot.
Drizzle each fig half with vinegar and fill each fig center with goat cheese. Sprinkle a couple flakes of crushed red pepper, if using, on top of the goat cheese and wrap each fig with a slice of prosciutto. Paint the outside of each fig very lightly with olive oil.
Put the figs on a cooler spot of the preheated grill. The idea here is that we want to grill the figs gently to cook the fig all the way through but to still get the char on the outside. Grill the figs on all sides so the prosciutto is starting to become crispy and the fig feels soft when squeezed, about 5 to 6 minutes.
While the figs are grilling, in a large bowl, toss the arugula with a little extra-virgin olive oil, some balsamic vinegar and the Parmigiano-Reggiano. Taste, and add salt, if needed. Arrange on individual serving plates or a serving platter and top with the figs. Drizzle a couple of drops of balsamic on each fig and serve.