Heat the oven to 400°. Cut the tough ends off the asparagus and discard them. Cut the spears into 1-inch pieces. Put the asparagus on a baking sheet and toss with the oil and 1/4 teaspoon each of the salt and pepper. Roast until tender, about 10 minutes.
Meanwhile, put the vinegar in a small saucepan. Simmer until 3 tablespoons remain. Stir in the brown sugar and the remaining 1/4 teaspoon pepper. Remove from the heat.
Cook the penne in a large pot of boiling, salted water until just done, about 13 minutes. Drain the pasta and toss with the butter, vinegar, asparagus, Parmesan, chicken and the salt. Serve with additional Parmesan.