In a large skillet, heat the olive oil over medium heat. Add the sliced onions and cook, stirring occasionally, for about 15-20 minutes until they are caramelized and golden brown. Add the minced garlic and cook for an additional 1-2 minutes until fragrant. Remove the onions and garlic from the skillet and set aside.
In the same skillet, season the chicken breasts with salt, pepper, and thyme. Cook the chicken over medium heat for about 6-7 minutes on each side, or until cooked through and no longer pink in the center. Remove the chicken from the skillet and set aside.
Pour the chicken broth into the skillet, scraping up any browned bits from the bottom. Stir in the heavy cream, Dijon mustard, and Parmesan cheese. Bring the mixture to a simmer and cook for 3-4 minutes until slightly thickened.
Return the caramelized onions and chicken to the skillet, coating the chicken in the creamy sauce. Simmer for an additional 2-3 minutes to heat everything through.
Serve the chicken topped with the creamy onion sauce and garnish with chopped parsley.