Heat a medium pan over medium-high heat. Add the pine nuts and cook for 4 minutes, until lightly browned, stirring. Remove from heat and set aside.
In a large mortar, mash together the green garlic and salt. Add the pine nuts and the pea shoot leaves and continue mashing into a paste. Add the olive oil, cheese, and lemon juice, stirring until combined. Season with additional salt if needed. Transfer to a medium bowl and set aside.
Bring a large pot of salted water to a boil over high heat. Add the linguine and cook until it is al dente. Drain the pasta. Add the pasta to the bowl with the pesto and toss until combined.
Divide the linguine on individual plates. Top with a dallop of additional pesto if desired. Serve immediately.