Let the chicken come to room temperature on the counter for about 15-20 minutes.
In a small bowl, prepare the seasoning mix by combining salt, pepper, thyme, garlic powder and paprika.
Place self-rising flour in a shallow bowl. Set aside ½ teaspoon of the seasoning mix to use for the gravy later; whisk the rest of the seasoning mix into the flour.
In a separate shallow bowl, whisk together buttermilk and egg.
Dip each cutlet in the buttermilk mixture first, then dredge in the flour mixture. Place the breaded cutlets on a wire rack to rest for a few minutes while you heat the oil and preheat the oven to 200°F.
Pour oil into a large Dutch oven or deep cast iron skillet to a depth of about ½-inch. Warm over medium heat until it’s really hot (about 350°F). The oil is ready when you drop a little bit of water on it and the water sizzles and pops.
Working in batches so that you don’t overcrowd the pan, fry chicken in oil until golden brown on both sides (about 3-4 minutes per side). Transfer to a paper towel-lined tray, season with additional salt and pepper (if desired), and then place (uncovered) on a tray in a 200°F oven to keep warm.
Repeat with remaining chicken.
While the chicken stays warm in the oven, make the gravy.
Whisk together broth and milk in a large measuring cup; set aside.
Spoon 2 tablespoons of oil that the chicken was fried in into a medium skillet (you can substitute with melted butter in lieu of the oil, if you like). Over medium heat, sprinkle in the flour and cook while whisking, until the flour is browned, about 1-2 minutes. Very gradually, whisk in the broth and milk. Add the remaining ½ teaspoon of seasoning mix that you set aside earlier. Cook and stir until gravy thickens. Remove from heat, taste and add seasoning if necessary, then spoon over chicken.