In a medium bowl, mix the masarepa, salt and sugar with a fork. Add the warm water and mix with a spoon. Then using your hands form dough. Let it sit for 5 minutes.
Divide the mixture into 4 portions and form each portion into a ball by rolling between the palms of your hands. Reserve a small piece of dough and set aside.
Place a piece of parchment paper or plastic in the kitchen counter, place the ball of dough and cover with another piece of paper or plastic and with a flat pot cover or a cutting board flatten into rounds, about ¼ inch thick.
Heat the vegetable oil in a large heavy pot to 350° F. Add the arepas in the heated oil one by one, fry for 3 minutes turning over once, about half way trough.
Using a slotted spoon, carefully remove the arepas from the oil and drain on paper towels. When the arepa is cold enough to handle, with a sharp knife make an opening about 1 ½ inches long on the side of the arepa, cutting to the middle of the arepa without slicing all the way through, just like making a pocket.
Crack 1 egg into a small ramekin and carefully pour the egg into the arepa pocket. Take the small piece of the dough you had aside and close the arepa. Add the arepa filled with egg to the hot oil and fry one more time for about 4 minutes, to cook the egg.
With a slotted spoon, remove the arepas from the oil and drain on paper towel and serve immediately.