Prepare a grill or grill pan to a very high heat.
Pour the balsamic vinegar in a small saucepan and bring to a boil over medium-high heat. Once boiling, reduce to a simmer and cook until the vinegar is reduced by half and is like a thick syrup, about five minutes. Let cool.
Cut the watermelon and feta cheese into 8 squares (3x3 and ½ inch thick). Drizzle the watermelon with a little bit of the olive oil and season with a bit of salt and pepper. Grill VERY quickly, just getting a slight char, about ten seconds per side.
Toss the arugula with about a teaspoon or two of olive oil. Season with salt and pepper.
Place a watermelon square on a platter, then a slice of feta, followed by another square of watermelon. Top with ½ cup arugula and drizzle with the reduced balsamic vinegar. Squeeze some lemon juice over the top and serve.
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