Heat the butter and oil in a cast iron or heavy-bottomed pan over medium heat. Lightly salt and pepper each side of the chicken breasts.
Sauté the first side of the chicken for 7 minutes. Then, flip and sauté for another 7 minutes. Remove from the pan and set aside.
Add the onions to the pan and sauté until softened and slightly caramelized, about 5 minutes. Add in the garlic and sauté for 30 seconds.
Pour in the wine and allow it to simmer until it reduces by half. This should take about 5 minutes.
Add in the chicken broth, heavy cream, sundried tomatoes, Parmigiano Reggiano, salt, pepper, and (optional) red pepper flakes. Add the chicken back into the pan along with any juices.
Simmer over medium heat for 10 minutes or until the chicken reaches an internal temperature of 165 degrees F.
Notes
If your chicken is very uneven in thickness, place it between two sheets of plastic wrap or parchment paper and pound it out gently with a mallet or rolling pin just until it's more even.
If you don't ant to use wine, you can replace it with more chicken broth.