Combine basil, garlic, and nuts in a food processor and pulse until coarsely chopped. Add 1/2 cup of the oil and process until incorporated and a smooth consistency. Season with salt and pepper, to taste. Stir in grated cheese and remaining oil. Pulse to combine. Store in the fridge in an airtight container for up to a week. Alternately, you can freeze in an ice-cube tray. Spray an ice cube tray with some cooking spray, fill with pesto, Lightly drizzle oil over the top and freeze until just firm, about 2 hours. Pop into freezer bags for easy storage. When you need to use one, take one out of the bag and thaw and use to your liking.