Place the pine nuts, fresh basil, shallot, jalapeño, garlic, and chopped green onions in the bowl of a food processor. Add the lime juice followed by the olive oil and vinegar. Sprinkle in the salt and Parmesan cheese.
Pulse until the mixture is smooth and well combined, stopping to scrape down the sides as needed.
Add the chopped lettuce to a large mixing bowl followed by the cucumbers and green onions.
Pour the dressing on top, and gently toss to combine and coat everything with the dressing.
Serve immediately.
