Bring a large pot of salted water to a boil and cook the pasta until al dente. Reserve 1 cup of the pasta cooking water, then drain.
Season chicken with salt and pepper. Heat 1 tbsp olive oil and 1 tbsp butter in a skillet over medium-high heat. Sear chicken until browned and cooked through, 4–6 minutes. Transfer to a plate.
Add remaining olive oil and butter to the pan. Add mushrooms in a single layer and let brown undisturbed for 3–4 minutes, then stir and cook until caramelized. Season lightly with salt.
Add shallot and garlic to the mushrooms and cook 1 minute. Pour in white broth and scrape up browned bits; reduce 2 minutes.
Stir in cream and Dijon, simmer 3–4 minutes to thicken slightly. Gradually stir in Parmesan until smooth. If too thick, loosen with reserved pasta water.
Return chicken to the pan, add drained pasta, and toss to coat. Adjust seasoning, garnish with parsley, and serve with extra Parmesan.
Notes
Use leftover rotisserie chicken to save time—add at the end to warm through.
For a deeper mushroom flavor, mix cremini with shiitake or porcini.
If you prefer a lighter sauce, use half-and-half and reduce the cheese slightly.