Preheat oven to 425 degrees F.
If your chicken breasts are very thick, place them on a cutting board, cover with plastic wrap, and pound lightly with the flat side of a meat mallet until no more than ¾-inch thick. You can slice them in half, if needed, for serving size pieces.
Pat the chicken dry with paper towels and season both sides with the salt, pepper, and Italian seasoning.
Heat the oil in a large skillet or sauté pan over MEDIUM-HIGH heat. When very hot, add the chicken and sear until golden brown on both sides, about 3 minutes per side (don’t cook through). Remove the pan from the heat, transfer the chicken to a plate, and set it aside.
Add the butter to the pan and return it to LOW heat. When melted, add the garlic and cook until fragrant, a minute or two. Add the water, balsamic vinegar, honey, mustard, and pepper and whisk until combined. In a small bowl, combine the cornstarch with 1 tablespoon water and whisk it into the sauce. Simmer for another minute or two, until thickened. Remove from the heat.
Return the seared chicken and any accumulated juices to the skillet and spoon some of the sauce over each piece of chicken. Place the rosemary sprigs over the top.
Transfer to the oven and bake, uncovered, until the chicken is cooked through (165 degrees F), about 12 to 15 minutes. Remove the pan from the oven.
Let it rest for 5 to 10 minutes before serving (sauce will continue to thicken).
Serve chicken with the balsamic pan sauce drizzled over the top.