Stand the watermelon half cut side down on a cutting board and slice away the rind, leaving a solid block of melon. Turn the block on its side and cut it into 8 squares, roughly 3 by 3 inches and 1-inch thick.
Pour the vinegar into a small saucepan and bring to a simmer over medium-high heat. Cook until reduced to a thick syrup consistency. Set aside.
Heat a nonstick grill pan over medium-high heat. Drizzle just enough olive oil over watermelon slices to thinly coat and place on hot grill pan. Grill each side about 2 minutes until grill marks appear; transfer to a plate and season with salt.
To assemble the salads, put about 1/4 cup of baby arugula on a serving plate, followed by a grilled slice of watermelon in the center, and top with a tablespoon of crumbled cheese and another 1/4 cup arugula. Add another watermelon slice and another tablespoon of cheese. Repeat with remaining ingredients. Finish each salad with a very light drizzle of olive oil and balsamic syrup. Dust with black pepper and serve immediately.