Bring a large pot of lightly salted water to a boil. Add fettuccini and cook to desired doneness. While this occurs, heat 2 Tbsp. of the oil in a skillet over medium heat. Add garlic, mushrooms and asparagus and cook, stirring, until the mushrooms and asparagus are just tender, about 5 minutes. Add 3/4 cup of the pasta cooking water to the pan and bring to a simmer. Drain pasta and add it, the Parmesan cheese and the basil to the skillet. Season with salt and pepper and toss to combine. Divide pasta among 4 bowls, or spoon into a large serving bowl. Drizzle will remaining 2 Tbsp. of oil. Serve pasta with additional freshly grated Parmesan cheese at the table, if desired.