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