In a blender, place cilantro and chicken stock. Blend until combined into a green liquid, about 1-2 minutes. Set aside.
Heat Instant Pot to saute mode. When hot, add oil then onion and peppers. Stir to saute, about 4-5 minutes. Turn the Instant Pot off (hit Cancel).
In a small bowl, combine paprika, cumin and salt. Season both sides of the chicken breasts with this mixture. Save any leftover seasoning not used on chicken. Set both aside.
Pour the cilantro stock into the Instant Pot. Add rice, canned tomatoes, and remaining seasoning; stir to combine all.
Layer seasoned chicken on top of the rice mixture. Do not submerge.
Layer shredded carrots on top of chicken. Do not mix in.
Lock the lid of the Instant Pot in place and make sure the release knob is not open. Press "Manual" on the Instant Pot and set the timer for 22 minutes.
When Instant Pot is done cooking (it will beep), manually release the pressure by moving the pressure release knob to "Release".
Once pressure is released, open the top of the Instant Pot carefully. Use two forks to shred the chicken. Then add the lime juice and mix all contents together.
Dish out portion and garnish with chopped, fresh cilantro and more lime if desired.