Season the chicken with salt, garlic powder, onion powder, paprika and ground black pepper. Set aside.
For the barbecue sauce, set the pressure cooker on the sauté function. Add the ketchup, brown sugar, molasses, Worcestershire, chipotle chile, vinegar, and garlic. Stir in the paprika, black pepper, and dry mustard, and simmer for about 3 minutes. The mixture will slightly thicken. Add the chicken broth and stir to combine. Turn the saute function off.
Place the seasoned chicken into the Instant Pot. Cover, seal and cook on high pressure for 10 minutes for chicken breasts and 14 minutes for chicken thighs. If you are using a combination, then cook for 14 minutes. When the cooking is done, let the pressure release naturally for 10 minutes. Then open the valve and release the remaining pressure.
Open the lid and using kitchen tongs, transfer the chicken to a large bowl, leaving the sauce in the pressure cooker. Set the cooker on the saute function and simmer the sauce until it has thickened to taste (remembering that it will continue to thicken as it cools), 4 to 6 minutes. Meanwhile, shred the chicken. Return the chicken to the pressure cooker and toss to combine.