Preheat your oven to 375°F if baking, or preheat the grill to about 400°F if grilling.
In a bullet blender or a large mixing bowl, combine the cream cheese, 1 tablespoon Parmesan, minced garlic, 1/3 cup chicken broth, kosher salt, and black pepper until smooth and well blended.
Add the cubed chicken and uncooked pearl couscous to the mixture and gently stir to coat; the couscous will absorb some of the liquid.
Tear a 12-inch piece of heavy-duty foil and fold into a shallow pocket on a rimmed baking sheet or an oven/grill-safe pan.
Spoon the chicken and couscous mixture into the center of the foil. Bring the short ends up and crimp together, then fold and crimp the long sides to form a sealed packet.
Place the foil packet on the baking sheet; if grilling, place the sheet and packets on one side of the grill and reduce the burners under the packets to low, then close the lid.
Cook for 15 minutes. After cooking, remove the packets to the rimmed baking sheet and let rest for 5 minutes; check the chicken for doneness and, if needed, reseal and cook 5 minutes longer.
Carefully open the foil packets away from your face to avoid steam, stir in the remaining 1/4 cup chicken broth, fluff the couscous with a fork, and top with the remaining Parmesan and chopped basil before serving.
