Begin by bringing a large pot of salted water to a boil. Once boiling, add the penne and cook according to the package instructions. Three minutes before the pasta is done, introduce the snap peas, asparagus, and kale into the pot, stirring occasionally. Reserve 1/2 cup of the cooking water before draining the pasta and vegetables.
In a blender, combine the olive oil, lemon zest, lemon juice, 1 teaspoon of salt, freshly ground pepper, and all but 2 tablespoons of the chopped chives. Add 3 tablespoons of cold water and blend until smooth, scraping down the sides as necessary. Pour the mixture into a large bowl, then stir in the tarragon and drizzle with additional olive oil.
Transfer the cooked pasta and vegetables into the chive puree, along with 1/4 cup of the reserved cooking water and half of the grated cheese. Season with salt and pepper to taste. Toss thoroughly, adding more cooking water to adjust the consistency if needed. Serve the dish topped with the remaining cheese and chives.