Combine vinegar, brown sugar, salt, hot sauce, and black pepper in your Instant Pot's insert, stirring or whisking to combine. Add chicken breasts.
Close Instant Pot lid (making sure the "venting" steam release knob is set to sealing, not venting). Select Manual, High Pressure, 8 minutes. (It takes my 6-quart Instant Pot Duo60 about 7-8 minutes to build enough pressure to seal, and then begin the 8-minute cook time.)
When cook time is completed, allow Instant Pot to sit (natural release) for 10 minutes, and then manually vent to relieve any residual pressure, and open the Instant Pot.
Remove chicken from Instant Pot and shred. Stir about 3/4 cup cooking liquid into chicken. Discard the additional cooking liquid, or reserve it to pass at the table for those people who like saucier BBQ. (See note about using the additional liquid if making a larger batch.)
Serve on buns with coleslaw on top or as an accompaniment on the side, and pass hot sauce if desired.
Amount of chicken: Although this recipe calls for 1 1/2 pounds of chicken (which is a good amount for about 6 sandwiches), you can use up to about 3 to 3 1/2 pounds of chicken, if you'd like to make a larger batch. There should be about 1 3/4 cups of cooking liquid in the bottom of the Instant Pot after the chicken has cooked (even though you started with only about 1 cup) – and that's enough cooking liquid to stir into up to 3 1/2 pounds of chicken, without adjusting any other ingredient amounts. So, for the 1 1/2 pounds of chicken this recipe calls for, plan to use about 3/4 cup of the liquid – and adjust upwards from there, if using more chicken.