Place the breasts and thighs on a plate and pat dry with a paper towel. Season with salt and pepper.
In a large pot or Dutch oven, heat the oil over medium heat. Add the peaches and cook, stirring often until they are softened but do not brown, 5 to 7 minutes. Add the chicken pieces and stir to coat. Pour in 1/2 cup of water and barbecue sauce, hot sauce and lemon juice. Stir to combine, making sure the chicken pieces are covered completely(add more water if needed). Increase heat to medium high and bring to a boil, and then reduce heat to medium low. Simmer uncovered until the sauce thickens and the chicken is cooked through 12 to 15 minutes.
Remove the pot from heat. Transfer the chicken pieces to cutting board and use two forks to shred meat. Return the meat to the pot and stir to combine.
Serve the peachy pulled chicken in between The bun halves.
Nutritional info 294 calories,17g protein, 6g fat, 44g carbs, 3g fiber, 924 mg sodium. Based on 6 servings.