In the same pan of the eggs put a little oil (in case there are some egg traces sticked to the bottom of the pan you can make them evaporate by adding a little bit of water).
Fry the ginger in the oil for a few minutes, then add the carrot. If necessary add a little water to encourage the cooking. When the carrot starts to become soft, add the rest of the vegetables and the onion, cooking over medium heat. Salt lightly and add small quantities of water or oil if they become too dry, being careful to mix them often and keep them consistent enough.