For the chicken: Fill a large heavy-bottomed high-sided skillet with enough oil to come 1/3 of the way up the sides. Place over medium heat.
Set up a breading station by whisking together the eggs, buttermilk and hot sauce in 1 pie plate or baking dish, and stirring together the fish fry mix and Cajun seasoning in another. Dip each chicken breast in the egg mixture, and then press into the fish fry mixture, making sure it gets completely coated. Set aside while the oil is heating.
When the oil reaches about 365 degrees F, carefully place the chicken into the oil. Fry in 2 batches so that the pot does not get overcrowded. Fry until just cooked through, 6 to 8 minutes. Remove from the oil and drain on a tray lined with paper towels.
For the grits: In medium pot, heat the milk to a simmer over medium heat. Whisk in the butter. Whisk in the grits and continue to whisk until thickened. Mix in 1/2 cup cheese and continue to stir until the cheese is melted and incorporated. Remove from the heat, sprinkle each serving with 1 tablespoon of the remaining cheese and serve along with the fried chicken.