Preheat the KUDU grill to medium-high heat.
Add the pistachios, basil leaves, salt, pepper, and garlic clove in a food processor. 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 the consistency is a thin paste. Taste and adjust seasoning, then set aside.
In a bowl, mix the green beans with the olive oil.
Place the green beans directly on the KUDU grill grates and grill for about 4 to 5 minutes per side (8 to 10 minutes total) until charred and tender.
Remove the beans from the grill, pour them into a serving dish, toss with pesto and serve immediately.