Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs; add the peas, asparagus and kale to the pot during the last 3 minutes of cooking and stir occasionally. Reserve 1/2 cup cooking water, then drain the pasta and vegetables.
Meanwhile, combine the olive oil, all but 2 tablespoons chives, the lemon zest and juice, 1 teaspoon salt and a few grinds of pepper in a blender. Add 3 tablespoons cold water and pulse until smooth, scraping down the inside of the blender. Transfer to a large bowl. Stir in the tarragon and drizzle with olive oil.
Add the pasta and vegetables to the chive puree along with 1/4 cup of the reserved cooking water and half of the cheese; season with salt and pepper. Toss well to coat, adding more cooking water to loosen, if necessary. Serve topped with the remaining cheese and chives.