Pour the balsamic vinegar into a small pot and place over medium-high heat. Reduce until the vinegar reaches the consistency of syrup.
Meanwhile, slice the chicken breasts in half parallel to the cutting board to make four thin pieces of chicken. Season the chicken with salt and pepper.
Heat the oil in a cast iron or other skillet over medium-high heat until nearly smoking. Add the chicken pieces in a single layer and let brown, undisturbed, for 1 minute. Turn the breasts, turn the heat to low, cover the pan, and let cook for 10 minutes. After ten minutes, remove the chicken to a plate next to the stove.
Turn the heat back up to high and let the liquid in the chicken pan reduce until you can drag a spoon through and leave a trail. Turn the heat back down to medium and add the onions. Let the onions cook for 2 minutes, then add the garlic and cook for a further 30 seconds. Add the halved tomatoes to the pan and sauté for 2 minutes.
Arrange the chicken pieces on a serving platter. Top with the tomato and onion mixture. Drizzle the balsamic reduction over the top and garnish with fresh basil.
Notes
Store this Balsamic Chicken with Tomatoes in an airtight container in the refrigerator, where it will keep well for up to 3 days. Although freezing is possible, it's best to enjoy it fresh to maintain the best texture and flavor.To reheat, gently warm the chicken and tomatoes in a skillet over medium heat until heated through, or microwave on medium power for a few minutes. If using a microwave, cover the dish to retain moisture and ensure even heating.