Preheat the oven to 375 degrees F.
Place the chicken breasts in a baking dish, skin side up. Loosen the skin from the meat with your fingers, leaving 1 side attached. Cut the goat cheese into 1/2-inch-thick slices and place 1 or 2 slices plus 1/4 of the julienned sun-dried tomatoes under the skin of each chicken breast, with goat cheese on top to sun dried tomatoes. Pull the skin over as much of the meat as possible so the chicken won't dry out. Rub each piece with olive oil, then sprinkle with salt and pepper.
Bake the chicken for 35 to 40 minutes, until the skin is lightly browned and the chicken reached 165F internal temperature.
Remove from the oven. Let rest for 5 minutes and serve.