Preheat grill to medium-high heat. Meanwhile, put arugula into a medium mixing bowl. Add lemon juice and 1 tablespoon olive oil; toss and set aside. Brush watermelon steaks on both sides with olive oil and place them on heated grill. Grill with the lid up for 2 minutes a side or until grill marks appear.
To assemble, spread out prepared arugula onto a serving platter and arrange the grilled watermelon on top. Sprinkle cheese and mint evenly over watermelon and drizzle with balsamic glaze.