In a medium saucepan, cook the rice according to package directions.
Meanwhile, in a large nonstick skillet, heat 1 tbsp olive oil over medium-high heat. Add the chicken, season with salt and pepper and cook, stirring, until browned on the outside, 2-3 minutes. Transfer to a plate.
Add the remaining 1 tbsp olive oil, the carrots and bell pepper to the skillet and season generously. Stir in the pesto and cook, stirring continuously, for 30 seconds. Stir in the coconut milk and all but 3 tbsp mint. Lower the heat and simmer, stirring occasionally, until the carrots are tender, about 10 minutes.
Return the chicken to the pan and cook until heated through, 3-4 minutes. Season with salt and pepper.
Stir the remaining 3 tbsp mint into the rice. Serve the chicken and sauce with the rice and lime wedges on the side.