Work with one piece of meat at a time. Sprinkle both sides with kosher salt and black pepper, then place it in the flour mixture. Turn to coat. Place the meat into the milk/egg mixture, turning to coat. Finally, place it back in the flour and turn to coat (dry mixture/wet mixture/dry mixture). Place the breaded meat on the clean plate, then repeat with the remaining meat.
Heat the oil in a large skillet over medium heat. Add the butter. Drop in a few sprinkles of flour to make sure it's sufficiently hot. When the butter sizzles immediately, you know it's ready. (It should not brown right away, if it does, the fire is too hot.) Cook the meat, 3 pieces at a time, until the edges start to look golden brown, about 2 minutes each side. Remove the meat to a paper towel-lined plate and keep them warm by covering lightly with another plate or a sheet of foil. Repeat until all the meat is cooked.
After all the meat is fried, pour off the grease into a heatproof bowl. Without cleaning the skillet, return it to the stove over medium-low heat. Add 1/4 cup of the grease back to the skillet and allow it to heat up.
For the gravy: When the grease is hot, sprinkle the flour evenly over the grease. Using a whisk, mix the flour with the grease, creating a golden-brown paste. Add more flour if it looks overly greasy; add a little more grease if it becomes too pasty/clumpy. Keep cooking until the roux reaches a deep golden brown color.
Pour in the milk, whisking constantly. Add the seasoned salt and black pepper to taste and cook, whisking, until the gravy is smooth and thick, 5 to 10 minutes. Be prepared to add more milk if it becomes overly thick. Be sure to taste to make sure gravy is sufficiently seasoned.
Serve the meat next to a big side of mashed potatoes. Pour gravy over the whole shebang!