Combine the basil, garlic, and pine nuts in a food processor and pulse until coarsely chopped. Add 1/2 cup of the oil and process until fully incorporated and smooth. Season with salt and pepper.
If using immediately, add all the remaining oil and pulse until smooth. Transfer the pesto to a large serving bowl and mix in the cheese.
If freezing, transfer to an air-tight container and drizzle remaining oil over the top. Freeze for up to 3 months. Thaw and stir in cheese.