Heat the oven to 350°F and bring a large pot of salted water to a boil.
Heat the olive oil in a large skillet over medium heat. When it's hot, add the garlic and cook, stirring frequently, until fragrant, about 2 minutes. Add the mushrooms, sprinkle with salt, and continue cooking until they release their liquid and it evaporates, about 8 to 10 minutes.
While the mushrooms cook, put half of the walnuts on a baking sheet and toast them in the oven until just fragrant, about 6 minutes; set aside.
Add the pasta to the water and cook until just tender but not soft. Finely chop the remaining walnuts. Once the mushroom liquid has reduced, add the chopped walnuts and chicken to the pan. Drain the pasta, reserving 1 cup of the cooking water, and add the pasta to the skillet. Toss the pasta with the mushroom mixture and add cooking water as needed to moisten everything. Add the toasted walnuts and parsley, toss again, season with salt and pepper, and serve garnished with Parmesan. Squeeze lemon wedges over to serve.