Place the potatoes on a baking sheet; drizzle olive oil over potatoes and season with salt and pepper.
Place the chicken breasts in a baking dish. Pour 1/2 cup of balsamic vinegar over the breasts; season with salt and pepper. Cover with aluminum foil.
Place the potatoes in the preheated oven and bake for 10 minutes; place the dish with the chicken in the oven and cook both the potatoes an chicken another 20 minutes; flip both the potatoes and chicken; reduce the oven heat to 350 degrees F (175 degrees C). Bake another 20 minutes.
Pour 1/2 cup of balsamic vinegar into a small saucepan and place over medium heat. Cook until reduced to about 1/4 cup. Place the chicken breasts atop the potatoes; drizzle with the reduced balsamic vinegar to serve.