In a bowl, mix the cornmeal and salt with a fork. Add the water and mix with your hands for about a minute, or until you have a soft yet firm homogenous mixture; this is the arepa dough. Let it rest uncovered at room temperature for 15 minutes.
Divide the dough into 12 equal balls, leaving a small piece of dough aside for use later in the recipe. Lay a piece of plastic wrap on the work surface, place a ball of dough in the center, cover with another piece of plastic, and with a heavy pan or pot cover flatten it until it forms an even round about 1/6-inch thick.
Place the oil in a heavy, deep pot, and bring to a temperature of 325 F. Add the arepas one by one so that they don’t stick to each other; don’t put too many at the time. Cook for 1½ minutes on each side and remove them very carefully with a slotted spoon so they don’t crack. They should have puffed up in the oil. Drain over
paper towels on a rack with the inflated side up.
Take scissors or a knife and cut a 1-inch opening in the edge of the arepa; go all the way in to reach the pocket of air formed on one side of it. Crack a whole egg into a small espresso cup and pour it into the arepa. Close the opening with a piece of uncooked dough, and immediately place in the hot oil again. Fry for 2 more minutes on each side. Repeat with the remaining balls of dough.
Drain briefly on paper towels. Serve immediately, very hot.