Preheat the oven to 400 degrees. Rinse and dry your potatoes to remove any dirt. Place the potatoes (whole) on a lined baking sheet. Drizzle the olive oil over the potatoes and season with salt. Toss to coat.
Roast for about 40 minutes. The cooking time can vary, depending on how big your potatoes are. (*see notes) You should be able to easily pierce all the way the potatoes with a knife when they are done.
While the potatoes are roasting, prepare the chicken by removing all of the meat from the bones and shredding it up. Place the chicken in a large mixing bowl, and pour the barbecue sauce over the top. Toss until the chicken is completely coated. Add more barbecue sauce to get it to your desired sauciness.
When the potatoes are done, remove from the oven and let them cool for about 10 minutes so that you are able to handle them. Once cooled, cut them down the middle, but only going about halfway through the potato so that it stays intact. Use a fork to fluff up the inside of the potato, and add a small dab of butter on top to melt.
To assemble, spoon about a 1/4-1/3 cup of the chicken into the baked potato and top with crumbled goat cheese.