I tried to pan fry quail for several years. The breast never came out right because they have three slices, and the bones prohibited the flour from properly coating. A friend and I had an 840 quail year, so I had more than I knew what to do with them. I taught myself to butterfly the quail. I've show this to lots of people since, and it's the only way they cook their quail anymore.
The first and most important step is to cut off the ball and socket joints like this;
Next you run your scissors under the rib cage on each side.
Then you clip off the sternum.
Pull the bones over the front of the quail breast, and it's a done deal.
You have a butterflied quail breast which will pan fry in less than 3 minutes on each side. Use your pan drippings for cream gravy. You don't have to use an egg wash if you bread them 30 minutes or so before you fry them. The breading will get gummy, and they'll come out crisp.
I've deep fried them for a field trial crowd, but I serve those with french fries and store bought breaded okra.