In a food processor, combine the cooked asparagus, garlic, basil, pine nuts, and 1/2 teaspoon of the salt. With the motor running, pour in 1/2 tablespoon of the olive oil and process until finely chopped. Set aside.
In a large skillet, heat the olive oil over medium heat. Add the shallot and 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 and bring to a boil. Reduce heat and simmer until the liquid is thickened, about 5 minutes.
Meanwhile, cook the spaghetti according to the package directions until al dente. Drain the spaghetti and add it to the skillet. Add the Parmesan, salt, pepper, and basil and toss well to combine. Serve immediately.