Why does this chicken recipe result in a glutinous mess?
The reason for the flour being part of the breading process is to create a barrier between the meat and the breading, which will during the frying step allow for the formation of small steam pockets and a crisper crust. In a Wiener schnitzel this leads to the characteristic wrinkled surface. On the other hand, it’s the flour-egg mix that acts as „glue“ for the breadcrumbs or whatever you are using.

The amount of flour that is needed to achieve this is minimal. If you end up with flour in the egg, then I suspect you aren’t shaking off the excess flour. 

The proper procedure starts with dry meat. This ensures that only little flour actually sticks to the meat when you then shake your meat to make sure all extra flour falls off. In this way you prevent the formation of what you describe as “glutinous mess”.

Then you proceed with the egg (again, let the extra drip off) and breadcrumbs.

Shallow-fry and serve immediately to ensure the crust is still crisp.