For each flatbread: Roll out 1 dough ball to 1/2-inch thickness. Grill dough on high heat until crisp on both sides. Brush with 2 Tbsp. barbecue sauce.
Cut 5 asparagus spears into 2-inch pieces and toss with 1 tsp. barbecue sauce; set aside. Top each flatbread with 3/4 cup arugula, 1 oz. prosciutto slices, reserved asparagus and 1 egg.