Cook the spaghetti according to the package directions until al dente. Drain.
Meanwhile, place the asparagus in a pot of lightly salted boiling water. Cook for 3 to 4 minutes, until tender-crisp. Drain and run under cold water to stop cooking.
Cut the spears in half. Roughly chop the stalk bottoms; cut tops into 1-inch pieces. In a food processor, combine the bottoms, garlic, ¼ cup basil, pine nuts, and ½ teaspoon salt. With the motor running, add 1 tablespoon oil and process until finely chopped.
In a large skillet, heat the remaining tablespoon of oil over medium heat. Add the shallot; cook until softened, 4 to 5 minutes. Add the asparagus mixture and cook until fragrant, 1 to 2 minutes. Add the chicken broth and asparagus tops and bring to a boil. Reduce heat and simmer until liquid is thickened, about 5 minutes.
Tear the remaining basil leaves and add to the skillet, along with the pasta, Parmesan, pepper, and remaining salt. Toss well. Serve immediately.