Brown the chicken cutlets. Season both sides of the chicken breasts with ½ teaspoon kosher salt and ½ teaspoon black pepper. In a large skillet, heat 1 tablespoon of the butter with 1 tablespoon olive oil over medium-high. Add the chicken and brown until golden on each side, about 3 minutes. Transfer the chicken to a plate (the chicken will continue cooking in the sauce later).

Cook the onion and garlic until tender. To the same skillet, reduce the heat to medium and add the remaining tablespoon of butter. Add ½ red onion, thinly sliced, and 2 cloves pressed garlic, and cook until the onion is tender, stirring occasionally.

Add the balsamic sauce flavorings, rosemary, and tomatoes. Add ¼ cup balsamic vinegar, 1 tablespoon brown sugar, 2 cups cherry tomatoes, 1 ½ tablespoons fresh rosemary, and ¼ cup water or chicken broth. Cook until the tomatoes soften and the sauce reduces and thickens, stirring often, about 5 minutes.

Return the chicken to the skillet. Add the cooked chicken back to the skillet and cook for 3-4 minutes, spooning the sauce over the cutlets, until the chicken registers 160°F. Taste for seasoning, and serve with more minced fresh rosemary if desired.