When I first started to cook fried rice on my own, some fifty or more years ago, all I had to use was a large cast iron skillet. It was well seasoned but not quite well enough to scramble eggs. In the more traditional preparation of egg fried rice, the eggs are added early on and continue to cook with the other ingredients. On my first solo attempt, I had eggs stuck to the frying pan and burning before the dish was competed cooked. It was a mess, and a waste. “There had to be a better way,” I thought. What I decided to do was cook the eggs separate in the one small Teflon pan we did have and then add the already scrambled eggs at the end of the cooking of the fried rice. It turns out that I like the outcome better than scrambling the eggs in the main pan. The curds are lighter and fluffier, have a better texture and mouth feel.
I have a well seasoned wok now and scrambling the eggs is no problem and I do it that way sometimes when time is short, but I still prefer my expedient method I devised so long ago. If you prefer to do it the more traditional way, cook the vegetables until almost done. Push the vegetable up on the side of the wok. Pour the beaten eggs in the center and scramble fry. When the eggs are almost completely cooked, stir the eggs and vegetables together. Add the leftover meat, stir fry to heat through, then add the rice and stir fry until the rice just begins to brown.