Place the carrots, red pepper, mushrooms, chicken, curry paste, sugar and broth into a 6 quart Instant Pot®.
Lock the lid and close the pressure release valve. Pressure cook on High pressure, setting the timer to 4 minutes (timer will begin counting down once pressure is reached- it takes about 15 minutes). When done, press Cancel and use the quick release method to release the pressure.
Remove the chicken from the pot and stir in the coconut milk. Shred the chicken and return to the pot. Select the Saute setting and cook for 2 minutes or until hot. Stir in the lime juice and lime zest just before serving. Serve sprinkled with the cilantro.