Preheat your grill to medium heat (around 500 degrees F).
In the bowl of a food processor add the pistachios, basil leaves, salt, pepper, and garlic clove. Pulse until crumbly. Add the parmesan cheese, lemon zest and juice and pulse until it becomes fine crumbs, uniform in shape. Drizzle in olive oil (2-4 tablespoons) while pulsing until consistency is a a thin paste. Taste and adjust seasoning, then set aside.
Make a grill “pan” out of aluminum foil. Start with a large square of foil, about 1.5 square feet in size. Fold up the edges to make a square with 2” rim. Pinch the 4 corners together to secure the shape of a square.
Add green beans to the foil plan and toss with 1 tablespoon of olive oil.
Place foil pan on hot grill grates, close lid and cook for 9-11 minutes. Stir green beans frequently during cooking to prevent burning. Beans are done when they have grill marks and are crisp tender. Cook longer for very tender green beans.
Remove beans from grill, pour into a serving dish, and immediately toss with pesto. Serve immediately.