Combine the basil, garlic, and pine nuts in a food processor or blender and pulse until coarsely chopped. With the machine running, slowly add the olive oil and process until smooth. Add the cheese and pulse until combined. Season with salt and pepper, to taste.

In a medium bowl, combine potatoes and pesto. Mix until potatoes are well covered. Spray a large piece of aluminum foil with cooking spray. Add the potatoes and wrap in a foil packet. Wrap another piece of foil around the packet so it is sturdy enough to grill.

Preheat the grill to medium-high heat. Place the potato packet on the grill. Cook for 15 minutes and then flip the packet. Cook for an additional 15 minutes, or until potatoes are tender and slightly crisp. Carefully remove the potatoes from the packet and serve warm.