I start out my chicken and dumplings pretty much like you but I made my dumplings a little different.
A little bit before the chicken is done I ladle off some of the broth and put it aside to cool. Then I pour my flour out on my table and make it into a volcano shape.
In the center of the volcano shape I break a couple of eggs, add my salt & pepper and pour in some of the cooled broth. Then I start out mixing from the center of the pile with a fork and end up needing the mixture. After that I roll it out and cut it into about 2 inch squares. Drop them into boiling broth and stir.
MMmmmmm Now I want to go fix me some chicken and dumplings!!
TWO eggs per cup of flour, though? Sounds like a lot of egg there.