No flour gravy is going to be really runny....and then the egg is certainly not going to soak up the gravy. So your plate is going to be full of liquid. You are making an egg/sausage soup.
Put Gooch on the line.
I am going to be cooking with heavy cream and butter. It will thicken up just fine. Also, in case of emergency, I can use a little flour. Just trying to limit it as much as I can.
I do this with pan sauces all the time. It won't get as thick as you could get it if you are cooking gravy out of a pouch, but it will get thick enough to give me what I want as far as creaminess and will satisfy the flavor profile I want, maybe even better than traditional methods.
That is my focus in eating low carb. Figure out what flavor or texture I am actually looking for from something I crave, then recreate that as best I can without the carb component.