Bring a large pot of salted water to a boil. Add the pasta and cook, stirring often, until it's just tender, about 9 minutes.
Scoop out 1 cup of the pasta water and set aside, then stir in the asparagus and cook until both the penne and the asparagus are tender, about 2 minutes.
Drain well and return the asparagus and pasta to the pot along with 3/4 cup of the Parmigiano-Reggiano, cream, 1/2 cup of the reserved pasta water, and 1/2 tsp black pepper. Set over medium heat and cook, stirring for a couple of minutes so cheese melts; add more of the pasta water if mixture becomes dry.
Stir in the chives, lemon zest and the lemon juice. Season generously with salt, pepper and more lemon juice to taste.
Serve with a sprinkling of the remaining Parmigiano-Reggiano.