Place the chicken breasts in the Instant Pot insert in a single layer.
Sprinkle the garlic powder, onion powder, and paprika evenly over the chicken; season with kosher salt to taste.
Pour 1 cup chicken broth over the chicken, then add 2/3 cup of the BBQ sauce on top.
Secure the lid and set the valve to sealing. Select Manual/Pressure Cook on high and set for 15 minutes.
When the cook cycle ends, allow natural pressure release for 10 minutes, then carefully quick-release any remaining pressure.
Check the chicken with a meat thermometer to ensure it reached at least 165°F; if not, reseal and pressure-cook a few more minutes.
Remove the chicken to a plate or bowl and shred with two forks.
Return the shredded chicken to the Instant Pot, add the remaining 2/3 cup BBQ sauce, and toss to coat; adjust salt to taste before serving.