Remove the tough ends of the asparagus spears and chop into 1-inch pieces. Reserve the tips.
Heat 1 tablespoon oil in a large pot over medium-low heat and cook garlic until light golden, 1 to 2 minutes. Add asparagus pieces and cook over medium heat, stirring often, until softened, about 5 minutes. Stir in mushrooms and coat well with oil. Pour in wine and cook until liquid evaporates, about 4 minutes. Add reserved asparagus tips and cook for 2 minutes. Discard garlic and season sauce with salt and pepper.
In the meantime, fill a large pot with lightly salted water and bring to a rolling boil. Cook tagliatelle at a boil until tender yet firm to the bite, about 8 minutes. Drain, reserving some of the cooking water.
Add tagliatelle to pot with the sauce, toss, and cook everything over high heat for 30 seconds to 1 minute. Drizzle with remaining olive oil and add some cooking water to thin sauce. Distribute tagliatelle on serving plates and sprinkle with Parmesan cheese.