Preheat oven to 425 degrees.
In a large bowl, whisk together balsamic vinegar, honey, mustard, garlic, and salt and pepper to taste (I like 1/2 teaspoon salt and 1/4 teaspoon pepper).
Add chicken and toss to coat. Marinate at room temperature for 20 minutes or in the refrigerator for up to 1 hour. Remove from marinade (reserve marinade) and pat chicken dry with paper towels.
In a 12-inch oven-safe skillet over medium-high heat, heat oil until shimmering. Add chicken in a single layer, skin-side down. Cook until browned, about 2 to 3 minutes.
Flip chicken and add tomatoes. Cook until tomatoes have started to pop release their juices, about 5 minutes.
Add reserved marinade and rosemary sprigs. Bring to boil, then transfer uncovered to the oven.
Bake until the chicken reaches 165 degrees on an instant-read thermometer, about 20 minutes. Remove from oven and remove rosemary stems.