COOK IT.
Turn the grill on to 350 ° F
Spritz your green onions and garlic with high smoke point oil. Season with salt and pepper.
Place vegetables either directly on your cooking grids or use a stainless steel grill topper accessory to ensure nothing falls through.
Remove garlic and onion from the grill, and allow them to cool to room temperature.
Using a blender, or high powered immersion blender, add all ingredients except for the parmesan and oil.
Pulse and blend to a semi-smooth consistency and then slowly drizzle in the oil. You want to emulsify these two liquids, creating a homogeneous mixture.
Add parmesan, blend once more and taste. You may add more salt or pepper if you choose, but this will give you the complexity of a blended grilled fresh pesto.