Preheat oven to 375°F (190°C). Cook pasta according to package directions until al dente; drain and set aside.
In a large oven-safe skillet, heat olive oil over medium-high heat. Add chicken pieces and cook until golden and cooked through, about 5-7 minutes. Remove chicken and set aside.
In the same skillet, add onion and garlic; cook until softened, about 2 minutes. Add mushrooms and cook until they release their liquid and start to brown, about 5 minutes.
Sprinkle flour over the vegetables and stir for 1 minute. Gradually whisk in chicken broth and heavy cream, stirring constantly until the sauce thickens, about 3-4 minutes.
Reduce heat to low. Stir in Parmesan cheese, thyme, salt, and pepper. Return chicken to the skillet. Add cooked pasta and toss to coat evenly.
Top with mozzarella cheese. Transfer skillet to the oven and bake for 20-25 minutes, until bubbly and golden on top.
Remove from oven, let rest for 5 minutes, then garnish with parsley before serving.
Tips
For a lighter version, substitute half-and-half for heavy cream.
Store leftovers in an airtight container in the refrigerator for up to 3 days.
Reheat in the oven or microwave until warmed through.