Place the chicken breasts in the Instant Pot in a single layer.
Sprinkle the chicken evenly with garlic powder, onion powder, and paprika, then season with kosher salt to taste.
Pour 1 cup chicken broth over the chicken, then spoon 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 time finishes, allow natural pressure release for 10 minutes, then carefully quick-release any remaining pressure.
Check the internal temperature with a thermometer; it should read at least 165°F. If needed, cook a few minutes longer.
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 evenly. Adjust salt to taste and serve.