Season chicken breasts with salt and pepper.
Heat olive oil and butter in a large skillet over medium-high heat.
Add chicken and cook for 5–6 minutes per side until golden and cooked through (165°F internal temp). Remove and set aside.
In the same skillet, add mushrooms and sauté for 3–4 minutes until softened.
Add garlic and cook for 30 seconds until fragrant.
Pour in white wine, scraping up browned bits. Simmer for 2 minutes.
Reduce heat to medium, then whisk in heavy cream, Dijon mustard, and thyme. Simmer for 3–4 minutes until slightly thickened.
Return chicken to the skillet, spooning sauce over it. Heat for 2 minutes.
Garnish with fresh parsley and serve hot over rice, pasta, or mashed potatoes.