Place cream cheese, Feta, and half & half in a large mixing bowl and beat with a hand mixer or immersion blender until mixture is light and fluffy. Set aside.
Heat balsamic vinegar in a small saucepan over medium-high heat. Bring to a gentle boil, stirring occasionally, and cook until the liquid is reduced by one-half and starts to cling to the back of a spoon. Reduce heat to low and keep warm until ready to use.
Drizzle balsamic glaze onto a serving platter and carefully arrange half of the watermelon squares on top.
Transfer some of the whipped Feta mixture to a pastry bag fitted with a large tip (or use a plastic sandwich bag with one of the bottom corners removed). Pipe the whipped Feta onto each watermelon square on the platter and top each with some fresh basil. Refill the pastry or plastic bag as necessary.
Place another slice of watermelon on top of the basil and gently press down to secure. Top each appetizer with some of the remaining whipped Feta and chopped pistachios.
Serve immediately garnished with small sprigs of fresh basil and a light drizzle of the remaining balsamic glaze, if desired. Enjoy!