2Meanwhile, make the sauce. In a large frying pan over medium heat, melt 2 tablespoons of the butter. Add the onion, season with salt, and cook, stirring, until golden, about 5 minutes. Add the mushrooms and wine, then sprinkle with the thyme and a little more salt and pepper, and cook, stirring, until the mushrooms soften, about 8 minutes. Transfer the mixture to a bowl.
3Melt the remaining 4 tablespoons butter in the frying pan over medium heat. Whisk in the flour and cook, whisking, for 2 minutes. Slowly add the broth, whisking constantly, then stir in the milk. Let simmer for 5 minutes.Add the chicken, reserved mushroom mixture, and 1 cup Parmesan, and stir until the cheese is melted and the mixture is slightly thickened to create a sauce, about 4 minutes. Remove from the heat and cover to keep warm.
4Cook the pasta to al dente according to package directions, then drain.
5Add the pasta to the pan and stir and toss to coat the pasta with the sauce and warm it through, about 3 minutes. Serve with a little more Parmesan and some parsley sprinkled over the top.