Pat the chicken dry and season both sides with salt and pepper. Heat olive oil in a large skillet over medium-high heat. Add the chicken and cook until golden and cooked through, about 6-7 minutes per side. Transfer to a plate and set aside.
In the same skillet, melt the butter over medium heat. Add the mushrooms and cook, stirring occasionally, until they release their moisture and turn golden, about 5-6 minutes.
Add the garlic and cook for 30 seconds until fragrant. Sprinkle the flour over the mushrooms and stir to coat, cooking for 1 minute.
Pour in the chicken broth, scraping up any browned bits from the bottom of the pan. Bring to a simmer and cook until slightly reduced, about 2 minutes.
Reduce the heat to low and stir in the heavy cream, Parmesan, and thyme. Simmer gently until the sauce thickens, about 3-4 minutes. Taste and adjust salt and pepper.
Return the chicken to the skillet, spooning some sauce over the top. Simmer for 2 minutes to heat through. Garnish with parsley and serve.