Combine sauce ingredients in a saucepan. Bring to a boil and cook until cherries soften. Remove from heat and set aside.
Preheat the oven to 350 degrees.
In a medium bowl, combine the goat cheese, fresh herbs and heavy cream to create an herbed goat cheese mixture; set aside as this will be stuffed inside the chicken breast.
Prepare your chicken by creating a pocket for the stuffing. Using a sharp paring knife, make a slit into the side of the chicken breast, cutting along one edge of the chicken. The slit should be about 2-3 inches, depending on how large your chicken breasts are. To fill your chicken breasts, place 1/4 of the herbed goat cheese mixture inside the slit of each breast. Season the chicken breast with salt & pepper to taste and set aside.
Prepare your skillet by heating up the oil and butter on high heat. Carefully place the chicken in the skillet and gently brown on all sides. Place the browned chicken in a baking dish and carefully pour the cherry balsamic braising sauce around the chicken. Bake in the preheated oven for 30-35 minutes.