Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Pat the chicken breasts dry with paper towels and season liberally with salt, pepper, and half of the Italian seasoning. Place them in the prepared baking dish.
In a large skillet, heat the olive oil over medium heat. Add the sliced onions and cook, stirring occasionally, for 10-12 minutes until they are soft and golden brown (caramelized).
Stir in the minced garlic and chopped sun-dried tomatoes; cook for another minute until fragrant.
Pour in the balsamic vinegar and chicken broth. Scrape the bottom of the pan to release any flavorful browned bits. Simmer for 2-3 minutes until the liquid reduces slightly.
Reduce the heat to low and stir in the heavy cream and remaining Italian seasoning. Simmer gently for 2 minutes to thicken the sauce slightly. Taste and adjust salt and pepper if needed.
Pour the creamy onion and tomato mixture evenly over the chicken breasts in the baking dish.
Bake uncovered for 20-25 minutes, or until the chicken reaches an internal temperature of 165°F (74°C).
Remove from the oven and let the dish rest for 5 minutes. Garnish with fresh parsley before serving.