Heat 2 tablespoons olive oil in a large skillet over medium heat; cook and stir carrots, ginger, and garlic until carrots are slightly softened, about 5 minutes. Transfer carrot mixture to a bowl. Add remaining 2 tablespoons olive oil to the same skillet.
Cook and stir chicken in the hot olive oil until no longer pink in the center, about 10 minutes. Add carrot mixture, water, peanuts, orange juice, soy sauce, brown sugar, ketchup, and red pepper flakes to chicken; stir to combine. Cover and simmer until sugar has dissolved, about 5 minutes.
Remove about 1/4 cup sauce from the skillet; whisk in cornstarch until dissolved and sauce is smooth. Pour cornstarch mixture back into chicken and sauce; cook until sauce is slightly thickened, about 5 more minutes.