Bring a large pot of salted water to a boil. Cook pasta according to package directions until al dente. Drain and set aside, reserving 1/2 cup pasta water.
While pasta cooks, heat olive oil in a large skillet over medium-high heat. Season chicken pieces with salt and pepper. Cook chicken until golden and cooked through, about 5-6 minutes. Remove chicken from skillet and set aside.
In the same skillet, add mushrooms and cook until browned, about 4-5 minutes. Add garlic and cook for 1 minute until fragrant.
Pour in chicken broth, scraping up any browned bits from the bottom of the pan. Let simmer for 2 minutes until slightly reduced.
Reduce heat to low. Stir in heavy cream, Parmesan cheese, and dried thyme. Simmer gently for 2-3 minutes until sauce thickens slightly.
Return chicken to the skillet. Add cooked pasta and toss to coat, adding reserved pasta water a little at a time if sauce is too thick. Season with salt and pepper to taste.
Serve immediately, garnished with fresh parsley.
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 gently on the stovetop with a splash of milk or broth.
Add a pinch of red pepper flakes for a hint of heat.