Slice each chicken breast in half lengthwise to create thinner cutlets. Season both sides with half of the salt and pepper.
Heat the olive oil and 1 tablespoon of butter in a large skillet over medium-high heat.
Add the chicken cutlets and cook for about 3 minutes per side until golden brown. Transfer the chicken to a plate. It will finish cooking later.
Reduce the heat to medium and melt the remaining tablespoon of butter in the same skillet.
Add the sliced red onion and cook for 3 to 4 minutes until softened.
Stir in the minced garlic and cook for about 30 seconds until fragrant.
Pour in the balsamic vinegar and stir in the brown sugar or honey.
Add the cherry tomatoes, fresh rosemary, and chicken broth or water.
Simmer for about 5 minutes, stirring occasionally, until the tomatoes begin to burst and the sauce thickens slightly.
Return the chicken and any accumulated juices to the skillet.
Spoon the balsamic sauce over the chicken and continue cooking for 3 to 4 minutes, or until the chicken reaches an internal temperature of 160°F (71°C).
Taste and adjust seasoning if needed.
Garnish with additional fresh rosemary and serve immediately.