Turn on a multi-functional pressure cooker (such as Instant Pot(R)) and select Saute function. Heat oil until hot. Add chicken thighs and cook until browned, about 3 minutes on each side. Transfer chicken to a plate.
Pour chicken broth into the pot. Cook and scrape up brown bits with a wooden spoon to deglaze the pot, about 1 minute. Turn off Instant Pot(R). Add celery, 1/2 cup barbecue sauce, and browned chicken thighs to the pot. Stir to combine.
Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 8 minutes. Allow 10 to 15 minutes for pressure to build.
Set an oven rack about 6 inches from the heat source and preheat the oven's broiler to low.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Transfer chicken to a baking sheet and baste with remaining 1/2 cup barbecue sauce.
