Heat an iron pan or non stick pan and grease with little oil or ghee.
When the pan is hot enough, add the batter to the pan and using a ladle spread the batter in circular shape by making circles with the ladle. Wait for a few minutes so the batter will start to set.
Once the batter starts to set in (in a few seconds), crack the egg and add it to the center of the batter.
Using a small fork or spoon, gently mix the egg (without touching the batter as its getting cooked), and spread the egg mixture almost evenly across the dosa batter. It may slightly spread over the edges of the dosa batter over to the pan which is totally fine.
Sprinkle little salt evenly over the egg mixture (according to your preference but eggs need only very little salt) and also add some black pepper powder if you wish.
Wait for the egg mixture to set in and almost half cooked, which will take one to two minutes of time, so keep the heat in low at this time. As the egg is getting cooked, add the ghee or oil evenly on the dosa and the edges of the batter which will make it more flavorful and crispy and also helps to cook more easily.
Once the egg mixture is about half cooked, gently flip the dosa (start separating from the sides first, then moving towards the center), and cook the other side of the egg dosa for another 1 minute or until the egg is fully cooked. Serve the delicious hot egg dosa with any chutney or sambar as you like!