Combine the chili powder, cumin, smoked paprika, onion powder, garlic powder, salt, and pepper in a small dish. Rub 1 tablespoon of olive oil and the spice mixture on the chicken breasts, fully coating them.
Add the remaining olive oil to the Instant Pot. Press the Sauté button. Once the oil is heated, brown the chicken breasts, two at a time, on each side for 2 minutes.
Press the Cancel button. Add in the broth and 1/2 cup BBQ sauce and stir. Add in the chicken breasts.
Press the Manual button and set the timer for 10 minutes. Allow pressure to release naturally for 10 minutes.
Remove the chicken breasts from the Instant Pot. Brush with remaining BBQ sauce until coated to your liking.
Leftovers will keep covered in the fridge for up to 4 days and can be reheated in the oven at 350F or slice cold and serve with salads.
Allow the Instant Pot to naturally release. If you quick release it, the chicken may not be fully cooked through.