Place chicken breasts between two sheets of plastic wrap or parchment paper. Using a meat mallet or heavy skillet, gently pound the thicker end of each breast to create uniform 3/4-inch thickness throughout. Pat dry with paper towels.
In a medium bowl, whisk together balsamic vinegar, olive oil, minced garlic, Dijon mustard, honey, dried Italian herbs, salt, and pepper until fully emulsified.
Place chicken in a shallow dish or resealable plastic bag. Pour 2/3 of the marinade over the chicken, reserving 1/3 for later use as glaze. Ensure chicken is evenly coated. Marinate in refrigerator for 30 minutes to 4 hours (optimal: 2 hours).
Position rack in middle of oven and preheat to 425°F (218°C). Allow at least 20 minutes for thorough preheating.
Remove chicken from marinade, discarding used marinade. Place chicken on a lightly oiled baking dish or rimmed sheet pan. Bake for 15 minutes.
While chicken bakes, pour reserved marinade into a small saucepan. Bring to a simmer over medium heat and reduce by approximately one-third until slightly thickened, about 5-7 minutes. Remove from heat.
After the initial 15 minutes, brush chicken generously with the reduced glaze. Return to oven and bake an additional 7-10 minutes, until internal temperature reaches 165°F (74°C) at the thickest part.
Remove chicken from oven, loosely tent with foil, and allow to rest for 5 minutes to redistribute juices and complete carryover cooking.
Slice chicken against the grain, drizzle with any remaining glaze, and sprinkle with fresh rosemary if desired.