Place the arugula in a medium bowl. Add 1 teaspoon of the oil, the balsamic vinegar, salt and pepper and toss to coat.

On each of four of the slices of bread layer ¼ cup of the cheese, then some halved grapes, then 4 small or 2 large basil leaves and finally about 1/2 cup of the arugula. Top each with the remaining slices of bread and brush both the outsides of each sandwich lightly with the oil.

Heat a medium-sized heavy cast-iron skillet or other skillet over medium-high heat and spray with cooking spray. Place 2 sandwiches in the skillet and weigh them down with another heavy skillet with a full can placed on top or a panini weight. Toast until the cheese begins to melt and the bread is browned and toasted on the edges, 3 to 4 minutes. Flip the sandwiches and toast for an additional 2 to 3 minutes. Remove from the pan and slice in half; repeat with the remaining 2 sandwiches.