Bring a large pot of generously salted water to a boil. Add the linguine pasta and cook according to the package instructions until al dente.
While the pasta cooks, prepare the pesto. In a food processor, combine the fresh watercress, fresh basil leaves, Parmesan cheese, toasted pine nuts, garlic cloves, and lemon juice. Pulse until the mixture is coarsely chopped.
With the food processor running on a low setting, slowly drizzle in the extra-virgin olive oil until the pesto becomes a smooth, emulsified sauce. Taste and season with kosher salt and black pepper.
Drain the linguine, making sure to reserve at least half a cup of the starchy pasta water. Return the drained pasta to the pot or place it in a large serving bowl.
Add the watercress pesto to the warm linguine. Toss gently to coat the noodles evenly. If the sauce seems too thick, splash in the reserved pasta water a little at a time until the desired silky consistency is reached.
Divide the pasta among serving bowls. Garnish with extra grated Parmesan cheese and a few fresh basil or watercress leaves. Serve immediately.