Fill a cast iron skillet or heavy-bottomed skillet about 1" thick with canola oil. Heat over medium high heat. It's ready when you can sprinkle a tiny bit of flour across it and it sizzles away immediately.
Whisk together the eggs and milk in a shallow bowl. Whisk together the flour and salt and pepper in a separate shallow bowl.
Dredge the pieces of meat in the flour, then egg. Repeat the process once more.
Nestle meat in the skillet and let cook on each side until it is a deep golden brown, about 6 minutes for each side. Once done, place on a cooling rack lined with paper towels. Repeat the process until all of the tenderized cube steak is gone.
For the gravy, drain off all of the oil and leftover bits of flour. If a lot of sediment is still leftover, wipe most of it out with a paper towel. Add 1/4 cup of oil back to the skillet and turn up to medium heat.
Whisk in the 1/2 cup of flour in increments, whisking as you go. Once the roux reaches a deep, almost golden color, add the milk in increments, whisking as you go. Continue whisking for 7-8 minutes over medium heat, until it thickens up and the lumps go away. Some lumps may remain, but that's okay. Season with salt and pepper.
Serve the chicken fried steak warm, with gravy poured on top.