In a medium-sized pot, bring the pomegranate juice to a boil over medium heat. Stir in the cornstarch and simmer until it thickens. Remove from heat and set aside.
Set the grill temperature to high and let it heat up. Remove the rind. Cut melon in 4 pieces roughly 2-inch by 4-inch rectangles. Brush the watermelon with olive oil and put on the grill for 2 minutes, turning to get grill marks on each side. Salt the watermelon when it comes off the grill. Add the arugula to a medium bowl and sprinkle with olive oil and salt and pepper, to taste.
Paint some of the pomegranate syrup on each plate. Top with 4 pieces of grilled melon and 1/8 cup of the goat cheese. Add more of the arugula salad and sprinkle with the remaining cheese. Drizzle each salad with balsamic mint syrup and some cracked black pepper. Sprinkle pomegranate seeds around each plate and serve.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.