Season the chicken on both sides with cajun seasoning and salt.
Heat a saucepan over medium heat. Once hot, add the oil (reserve 1/3 for the chicken). Saute the garlic and onion until soft. Stir in the broth and bring to a boil. Stir in the grits, cooking according to the instructions on the packaging (may need to turn down the heat). Once creamy remove from the heat and stir in the scallions and butter, season with salt and pepper to taste. Set aside with a lid to keep warm.
Heat an oven safe sauté pan over medium heat. Add the reserved oil. Brown the chicken nicely on both sides to golden brown. Place the pan in the oven and cook until the chicken reaches an internal temperature of 160 degrees using a thermometer. Then set aside on a cutting board for 3 to 5 minutes before slicing.
Place the pan over medium heat and deglaze the pan with chicken broth. Scraping up anything stuck on the bottom of the pan. Stir in the lemon zest and thyme. Turn to a slow simmer to build flavor. Once flavorful, slowly add the flour water slurry a little at a time to thicken the sauce to a creamy consistency. Season with salt and pepper to taste.
Serve the chicken with the grits. Spoon the sauce over top. Garnish with scallions.