Pat chicken dry and season on all sides with salt and pepper. Place the flour in a shallow dish and season with salt and pepper. Dredge each chicken cutlet in the seasoned flour, shaking off any excess flour.
In a large skillet, heat olive oil over medium-high heat. Stir in butter. When the butter is foamy, add the chicken cutlets to the pan. Cook until the chicken is browned on both sides and the meat reaches an internal temperature of 165° F, about 3-4 minutes per side. Do not overcrowd the pan, so brown the chicken in batches if necessary. Remove the chicken to a plate and set aside while you make the sauce.
Add the sliced onion and apples to the skillet and cook over medium-high heat until the onion is translucent, about 5 minutes. Add the garlic and rosemary; cook, stirring constantly, for 30 seconds.
Add the chicken broth to the skillet; use a wooden spoon to scrape up any browned bits from the bottom of the pan. Simmer for 3-4 minutes, so that the liquid reduces by about half.
Reduce heat to low, gradually whisk in the cream, Dijon and brown sugar. start with about ¼ cup of cream, and then slowly add more, if desired. Cook for a few more minutes to let the sauce thicken (do not boil). Taste and season with salt and pepper.
Return chicken to the skillet; garnish with additional fresh herbs. Serve immediately.