Heat 1 teaspoon of oil in a pan over medium high heat. Add the carrots and cook for 1-2 minutes. Add 2 tablespoons of water and cook, stirring occasionally, for 4-6 minutes until the water has cooked off and carrots have started to soften.
Add the bok choy to the pan and cook for 3-4 minutes more, until both bok choy and carrots are cooked through. Season the vegetables with salt and pepper to taste, then transfer to a plate and cover.
Wipe out the pan with a paper towel.
Heat the remaining 2 teaspoons of oil over high heat. Season the chicken generously and add to the pan. Cook, stirring occasionally, for 7-8 minutes or until chicken is browned and cooked through.
While the chicken is cooking, make the peanut sauce. Place the ginger, garlic, peanut butter, lime juice, brown sugar, soy sauce, red pepper flakes and 2 tablespoons of water in a food processor; blend until smooth. Add more water to further thin the sauce if desired.
Return the vegetables to the pan with the chicken and cook until heated through, 1-2 minutes. Drizzle the peanut sauce over the top of the chicken and vegetables. Top with chopped peanuts and serve.