Fill a large bowl with ice water.

Bring a small saucepan of water to a boil over medium-high heat. Working in batches if needed, add basil and spinach; cook, stirring occasionally, until vibrant green and wilted, 30 seconds to 1 minute. Remove using a slotted spoon, and immediately transfer to ice water bath. Let stand until cooled, about 30 seconds. Drain well; thoroughly squeeze dry.

In the work bowl of a small food processor, pulse pine nuts, garlic, and lemon zest and juice until coarsely ground. Add cheese, oil, salt, and pepper; pulse until combined. Add drained basil and spinach, and pulse until finely chopped and well combined, stopping to scrape sides of bowl. Refrigerate in an airtight container for up to 1 week.