Preheat a grill or grilll pan on medium heat. Lightly brush the pita with olive oil, then grill until marked on the bottom, about 2 minutes. Flip the bread and immediately top with the fontina. Continue grilling until the cheese melts, about 2 minutes. Transfer to a platter and top with the prosciutto.
Combine the arugula, parsley, grape tomatoes, and red onion in a large bowl. Drizzle with remaining olive oil; season with salt and pepper and toss. Pile on top of the pita; sprinkle with blue cheese and drizzle with balsamic glaze. Cut into quarters and serve with chips.