Heat a large pan over medium-high heat. Add 1 Tablespoon of olive oil to the pan and when heated, add the grated carrots and a little salt and pepper. Combine and cook for 1 minute.
Next, add the rice and stir to coat. Pour 2 ½ cups of the stock in the pan. Bring the stock to a boil, cover, and then reduce the heat to a simmer. Cook for 15 to 18 minutes, or until the rice is tender and the liquid has been absorbed. Set aside.
Chop the breasts into bite-size pieces. Combine 2 Tablespoons of the olive oil, ground thyme, lime zest, salt, and pepper in a shallow dish and then add it to the skillet.
Add the chopped breasts to the skillet and saute with the sauce until the chicken is cooked all the way through. Transfer to a plate and set aside.
Return the skillet to the stovetop and set to medium-high heat. Add the remaining Tablespoon of olive oil. Once it's heated, add the garlic, red pepper flakes, and coriander. Cook, stirring constantly, for about 1 minute. Lastly, add the lime juice, parsley, butter, scallions, and the remaining ½ cup of stock and continue to cook for about 3 minutes.
As the sauce cooks, it will start to thicken. Continue to stir the sauce until it thickens, another 1-2 minutes on medium heat.
Divide the carrot rice among 4 serving bowls. Place the cooked skillet chicken onto the bed of rice in each bowl and then pour the garlic butter sauce over the pieces of meat. Top with additional scallions. Serve immediately.