Warm the olive oil in a large skillet over medium-high heat until it's hot and shimmering.
Season the chicken breasts well on both sides with salt and pepper.
Place the chicken in the skillet and cook for about 6 to 7 minutes on each side until they turn golden and are cooked through (internal temperature should reach 165°F). Remove from skillet and keep warm.
Add the sliced onion to the same pan and stir occasionally, cooking them for 3 to 4 minutes until tender.
Stir in the minced garlic and continue cooking for another 1 to 2 minutes until the aroma develops.
Pour in the chicken broth, scraping the bottom of the pan to loosen any browned bits for extra flavor.
Mix in the heavy cream, Dijon mustard, and dried thyme, stirring everything together well.
Return the chicken breasts to the skillet and ladle the sauce over them.
Turn the heat to low, cover the pan, and let it simmer gently for 5 to 10 minutes to thicken the sauce.
Sprinkle the dish with freshly chopped parsley before serving right away.